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Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems

J
jfrchicanog

This document describes research on developing an efficient hill climber algorithm for constrained pseudo-Boolean optimization problems. It discusses how scores can be computed to identify improving moves in the search space and updated efficiently as the solution changes. The key ideas are to compute scores initially and then only update a small, constant number of scores after each move instead of recomputing all possible scores from scratch. This approach is extended to handle multi-objective optimization problems with constraints by considering both strong and weak improving moves.

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1 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems Francisco Chicano, Darrell Whitley, Renato Tinós
2 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i S1( r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i • Considering binary strings of length n and Hamming distance… Solutions in a ball of radius r r=1 r=2 r=3 Ball of radius r Previous work Research Question How many solutions at Hamming distance r? If r << n : Θ (nr)
3 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • We want to find improving moves in a ball of radius r around solution x • What is the computational cost of this exploration? • By complete enumeration: O (nr) if the fitness evaluation is O(1) • Previous work proposed a way to find improving moves in ball of radius r in O(1) (constant time independent of n): Szeider (DO 2011), Whitley and Chen (GECCO 2012), Chen et al. (GECCO 2013), Chicano et al. (GECCO 2014) • The approach was extended to the multi-objective case: Chicano et al. (EvoCOP 2016) • All the previous results are for unconstrained optimization problems Improving moves in a ball of radius r r Ball of radius r Previous work Research Question
4 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Research Question Ball of radius r Previous work Research Question
5 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Definition: • where f(i) only depends on k variables (k-bounded epistasis) • We will also assume that the variables are arguments of at most c subfunctions • Example (m=4, n=4, k=2): • Are Mk Landscapes a small subset of problems? Are they interesting? • Max-kSAT is a k-bounded pseudo-Boolean optimization problem • NK-landscapes is a (K+1)-bounded pseudo-Boolean optimization problem • Any compressible pseudo-Boolean function can be reduced to a quadratic pseudo-Boolean function (e.g., Rosenberg, 1975) Mk Landscape (Whitley, GECCO2015: 927-934) Pseudo-Boolean functions Scores Update Decomposition The family of k-bounded pseudo-Boolean Optimization problems have also been described as an embedded landscape. An embedded landscape [3] with bounded epistasis k is de- fined as a function f(x) that can be written as the sum of m subfunctions, each one depending at most on k input variables. That is: f(x) = mX i=1 f(i) (x), (1) where the subfunctions f(i) depend only on k components of x. Embedded Landscapes generalize NK-landscapes and the MAX-kSAT problem. We will consider in this paper that the number of subfunctions is linear in n, that is m 2 O(n). For NK-landscapes m = n and is a common assumption in MAX-kSAT that m 2 O(n). 3. SCORES IN THE HAMMING BALL For v, x 2 Bn , and a pseudo-Boolean function f : Bn ! R, we denote the Score of x with respect to move v as Sv(x), defined as follows:1 Sv(x) = f(x v) f(x), (2) 1 We omit the function f in Sv(x) to simplify the notation. S(l) v (x) = Equation (5) cl change in the mov f(l) the Score of th this subfunction w On the other hand we only need to c changed variables acterized by the m we can write (3) a S 3.1 Scores De The Score value tion than just the c in that ball. Let us balls of radius r = xj are two variabl ments of any subfu f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x) x1 x2 x3 x4
6 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Based on Mk Landscapes Going Multi-Objective: Vector Mk Landscape Pseudo-Boolean functions Scores Update Decomposition x1 x2 x3 x4 x5 f (1) 1 f (2) 1 f (3) 1 f (4) 1 f (5) 1 f (1) 2 f (2) 2 f (3) 2 (a) Vector Mk Landscape x3x4x5 f1 f2
7 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Going Multi-Objective & Constrained Pseudo-Boolean functions Scores Update Decomposition f1 f2 x1 x2 x3 x4 x5 f (1) 1 f (1) 2 f (2) 1 f (2) 2 f (3) 1 g (1) 1 g (2) 1 g (3) 1g1 Feasible solutions: . BACKGROUND In constrained multi-objective optimization, there is a vec- r function f : Bn ! Rd to optimize, called the objective nction. We will assume, without loss of generality, that l the objectives (components of the vector function) are be maximized. The constraints of the problem will be ven in the form1 g(x) 0, where g : Bn ! Rb is a vec- r function, that will be called constraint function2 . That , a solution is feasible if all the components of the vector nction g are nonnegative when evaluated in that solution. his type of constraints does not represent a limitation. Any her equality or inequality constraint can be expressed in he form gi(x) 0, including those that use strict inequality onstraints (> and <)3 . The set of feasible solutions of a roblem will be denoted by Xg = {x 2 Bn |g(x) 0}. Given a vector function f : Bn ! Rd , we say that solution Objectivefunction Constraint function
8 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Let us represent a potential move of the current solution with a binary vector v having 1s in the positions that should be flipped • The score of move v for solution x is the difference in the fitness value of the neighboring and the current solution • Scores are useful to identify improving moves: if Sv(x) > 0, v is an improving move Scores: definition Current solution, x Neighboring solution, y Move, v 01110101010101001 01111011010101001 00001110000000000 01110101010101001 00110101110101111 01000000100000110 01110101010101001 01000101010101001 00110000000000000 Pseudo-Boolean functions Scores Update Decomposition vector Mk Landscape of Figure 1(a). move in Bn can be characterized by a binary n having 1 in all the bits that change in the so Score of a move, has been previously defined ment in the objective function when that move is finition 4 (Score). For v, x 2 Bn , and a on f : Bn ! Rd , we denote the Score of x with ve v for function f as S (f) v (x), defined as follow S(f) v (x) = f(x v) f(x), is the exclusive OR bitwise operation.
9 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) Scores update r Selected improving move Update the scores Pseudo-Boolean functions Scores Update Decomposition
10 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) • How can we do it less expensive? • We have still O(nr) scores to update! • … thanks to two key facts: • We don’t need all the O(nr) scores to have complete information of the influence of a move in the objective or constraint (vector) function, only O(n) scores • From the ones we need, we only have to update a constant number of them and we can do each update in constant time Key facts for efficient scores update r Pseudo-Boolean functions Scores Update Decomposition
11 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work What is an improving move in MO? • An improving move is one that provides a solution dominating the current one • We define strong and weak improving moves. • We are interested in strong improving moves Taking improving moves Taking feasible moves Algorithm f2 f1 Assuming maximization Can we safely take only strong improving moves?
12 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work We need to take weak improving moves • We could miss some higher order strong improving moves if we don’t take weak improving moves S2 S1 stored stored not stored current solution x1 x2 x3 x4 x5 f2 f2 g (1) 1 g (2) 1 g (3) 1 S (f) v1[v2 (x) = S(f) v1 (x) + S(f) v2 (x) Taking improving moves Taking feasible moves Algorithm
13 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Cycling • We can make the hill climber to cycle if we take weak improving moves S2 S1 stored current solution S2 S1 stored current solution Taking improving moves Taking feasible moves Algorithm
14 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Solution: weights for the scores • Given a weight vector w with wi > 0 S2 S1 w 1st strong improving 2nd w-improving 2nd w-improving A w-improving move is one with w · Sv(x) > 0 Taking improving moves Taking feasible moves Algorithm
15 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility does not only depends on the scores • In order to classify the moves as feasible or unfeasible we have to check all the scores, even if they have not changed • This can be done in O(n), but not in O(1) in general g2 g1 Unfeasible solution current solution v1 v1 v2 Feasible region Taking improving moves Taking feasible moves Algorithm
16 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility in the Hamming Ball x3 x4 x5 f (2) 1 f (2) 2 f (3) 1 g (2) 1 g (3) 1 S (f) v1[v2 (x) = S(f) v1 (x) + S(f) v2 (x) (1) S (g) v1[v2 (x) = S(g) v1 (x) + S(g) v2 (x) (2) g2 g1 Unfeasible stored solution current solution v1 v1 v2 Feasible region v2 Unfeasible stored solution Feasible not stored solution Taking improving moves Taking feasible moves Algorithm
17 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility in the Hamming Ball g2 g1 current solution v1 v1 v2 Feasible region v2 w* w*-feasible • Bad news: this does not work when the current solution is unfeasible • Worse news: there is no “efficient” way to identify a feasible solution when the current solution is unfeasible Taking improving moves Taking feasible moves Algorithm
18 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Unfeasible regionFeasible region No w-improving or w*-feasible moves ▶ stop strong improving in g w*-improving in g solution y otherwise Hill Climber Taking improving moves Taking feasible moves Algorithm
19 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work MNK Landscape Problem Results Source Code Why NKq and not NK? Floating point precision • An MNK Landscape is a multi-objective pseudo-Boolean problem where each objective is an NK Landscape (Aguirre, Tanaka, CEC 2004: 196-203) • An NK-landscape is a pseudo-Boolean optimization problem with objective function: where each subfunction f(l) depends on variable xl and K other variables • The subfunctions are randomly generated and the values are taken in the range [0,1] • In NKq-landscapes the subfunctions take integer values in the range [0,q-1] • We used shifted NKq-landscapes in the experiments with values in two ranges: • [-49, 50] slightly constrained problems (2% unfeasible) • [-50, 49] highly constrained problems (80% unfeasible) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) f(x) = NX l=1 f(l) (x) 1 • In the adjacent model the variables are consecutive f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
20 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work 0 2000 4000 6000 8000 10000 12000 14000 16000 1 2 3 4 5 6 7 8 9 10 Averagetimepermove(microseconds) N (number of variables in thousands) r=1 r=2 r=3 Runtime: highly constrained MNK Landscapes Neighborhood size: 166 billion Problem Results Source Code
21 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work ries from 1 to 3. These instances are slightly constrained th around 2% of the search space being unfeasible10 . We rformed 30 independent runs of the algorithm for each nfiguration, and the results are the average of these 30 ns. 0 200 400 600 800 1000 1200 1400 1600 10 20 30 40 50 60 70 80 90 100 Averagetimepermove(microseconds) N (number of variables in thousands) r=1, d=1, b=1 r=1, d=2, b=1 r=1, d=1, b=2 r=1, d=2, b=2 r=2, d=1, b=1 r=2, d=2, b=1 r=2, d=1, b=2 r=2, d=2, b=2 r=3, d=1, b=1 r=3, d=2, b=1 r=3, d=1, b=2 r=3, d=2, b=2 gure 3: Average time per move in microsec- Figure 4 for the rithm 1 fo b = 1, N [ 50, 49], In this c the time p per move Figure 3, times sma the unfeas all the mo time. 5.2 Qu Runtime: slightly constrained MNK Landscapes Neighborhood size: 166 trillion Problem Results Source Code from 1 to 3. These instances are slightly constrained round 2% of the search space being unfeasible10 . We med 30 independent runs of the algorithm for each uration, and the results are the average of these 30 0 200 400 600 800 1000 1200 1400 1600 10 20 30 40 50 60 70 80 90 100 N (number of variables in thousands) r=1, d=1, b=1 r=1, d=2, b=1 r=1, d=1, b=2 r=1, d=2, b=2 r=2, d=1, b=1 r=2, d=2, b=1 r=2, d=1, b=2 r=2, d=2, b=2 r=3, d=1, b=1 r=3, d=2, b=1 r=3, d=1, b=2 r=3, d=2, b=2 0 1 Figure 4: A for the Mul rithm 1 for co b = 1, N = 1 [ 50, 49], and In this case, the time per m per move in th Figure 3, even times smaller) the unfeasible all the moves time. 5.2 Qualit
22 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Source Code https://github.com/jfrchicanog/EfficientHillClimbers/tree/constrained-multiobjective Problem Results Source Code
23 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Conclusions and Future Work Conclusions & Future Work • Adding constrains to the MK Landscapes has a cost in terms of efficiency: from O(1) to O(n) • The space required to store the information is still linear in the size of the problem n Conclusions • Generalize to other search spaces • Combine with high-level algorithms (MOEA/D) Future Work
24 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Acknowledgements Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems
25 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Examples: 1 and 4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) f(1) f(2) f(3) f(4) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) Pseudo-Boolean functions Scores Update Decomposition
26 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Example: 1,4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 r Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) n i S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1(x) = f(x 1) f(x) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) Ball Pr i=1 n i S1(x) = Sv(x) = f(x v) f(x) = S4(x) = S1(x) = f(x 1) f(x) x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We don’t need to store S1,4(x) since can be computed from others If none of 1 and 4 are improving moves, 1,4 will not be an improving move Pseudo-Boolean functions Scores Update Decomposition
27 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Quality • 50% Empirical Attainment Functions (Knowles, ISDA 2005: 552-557) Problem Results Source Code 0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06 10 20 30 40 50 60 70 80 90 100 Averagequalityofbestfoundsolution N (number of variables in thousands) r=1, b=1 r=1, b=2 r=2, b=1 r=2, b=2 r=3, b=1 r=3, b=2 Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 (a) N = 10, 000, b = 1 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (b) N = 50, 000, b = 2 Figure 6: 50%-empirical attainment surfaces of the 30 runs of the Multi-Start Hill Climber based on Acknowledgements This research was partially funded by gram, the Spanish Ministry of Education, (CAS12/00274), the Spanish Ministry of E petitiveness and FEDER (TIN2014-573 sity of M´alaga, Andaluc´ıa Tech, the A Scientific Research, Air Force Materiel (FA9550-11-1-0088), the FAPESP (2015/ 7. REFERENCES [1] Hernan E. Aguirre and Kiyoshi Tan properties of multiobjective MNK-la Proceedings of CEC, volume 1, page [2] Wenxiang Chen, Darrell Whitley, D Adele Howe. Second order partial de NK-landscapes. In Proceeding of GE 503–510, New York, NY, USA, 2013 [3] Francisco Chicano, Darrell Whitley, Sutton. E cient identification of im ball for pseudo-boolean problems. In GECCO, pages 437–444. ACM, 201 [4] Francisco Chicano, Darrell Whitley, Tin´os. E cient hill climber for mult pseudo-boolean optimization. In Pro EvoCOP, pages 88–103, 2016. [5] Yves Crama, Pierre Hansen, and Br The basic algorithm for pseudo-boo revisited. Discrete Applied Mathema 29(2-3):171–185, 1990. [6] Brian W. Goldman and William F. optimization using the parameter-le pyramid. In Proceedings of GECCO 0 500000 1e+06 10 20 30 40 50 60 70 80 90 100 Averagequ N (number of variables in thousands) Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (FA9550- 7. RE [1] Hern prop Proc [2] Wen Ade NK- 503– [3] Fran Sutt ball GEC [4] Fran Tin´o pseu Evo [5] Yve The revi 0 500000 1e+06 10 20 30 40 50 60 70 80 90 100 Averagequ N (number of variables in thousands) Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (F 7. [ [ [ [ [ Single-objective constrained problem Bi-objective constrainedproblems 1 constraint 2 constraints
28 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Example: 1,2 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) f(1) f(2) f(3) x1 x2 x3 x4 f(1) f(2) f(3) x1 x2 x3 x4 f(1) x1 x2 Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) x1 and x2 “interact” Pseudo-Boolean functions Scores Update Decomposition
29 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Decomposition rule for scores • When can we decompose a score as the sum of lower order scores? • … when the variables in the move can be partitioned in subsets of variables that DON’T interact • Let us define the Co-occurrence Graph f(1) f(2) f(3) f(4) x1 x2 x3 x4 There is an edge between two variables if there exists a function that depends on both variables (they “interact”) x4 x3 x1 x2 Pseudo-Boolean functions Scores Update Decomposition
30 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Whitley and Chen proposed an O(1) approximated steepest descent for MAX-kSAT and NK-landscapes based on Walsh decomposition • For k-bounded pseudo-Boolean functions its complexity is O(k2 2k) • Chen, Whitley, Hains and Howe reduced the time required to identify improving moves to O(k3) using partial derivatives • Szeider proved that the exploration of a ball of radius r in MAX-kSAT and kSAT can be done in O(n) if each variable appears in a bounded number of clauses Previous work Ball of radius r Improving moves Previous work Research Question D. Whitley and W. Chen. Constant time steepest descent local search with lookahead for NK-landscapes and MAX-kSAT. GECCO 2012: 1357–1364 W. Chen, D. Whitley, D. Hains, and A. Howe. Second order partial derivatives for NK-landscapes. GECCO 2013: 503–510 S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139–145, 2011
31 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to store • In terms of the VIG a score can be decomposed if the subgraph containing the variables in the move is NOT connected • The number of these scores (up to radius r) is O((3kc)r n) • Details of the proof in the paper • With a linear amount of information we can explore a ball of radius r containing O(nr) solutions x4 x3 x1 x2 x4 x3 x1 x2 S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f S1,2(x) 6= S1(x) + S2(x) l=1 l=1 S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We need to store the scores of moves whose variables form a connected subgraph of the VIG Pseudo-Boolean functions Scores Update Decomposition
32 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to update • Let us assume that x4 is flipped • Which scores do we need to update? • Those that need to evaluate f(3) and f(4) f(1) f(2) f(3) f(4) x1 x2 x3 x4 x4 x3 x1 x2 • The scores of moves containing variables adjacent or equal to x4 in the VIG Main idea Decomposition of scores Constant time update
33 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to update and time required • The number of neighbors of a variable in the VIG is bounded by c k • The number of stored scores in which a variable appears is the number of spanning trees of size less than or equal to r with the variable at the root • This number is constant • The update of each score implies evaluating a constant number of functions that depend on at most k variables (constant), so it requires constant time x4 x3 x1 x2 f(1) f(2) f(3) f(4) x1 x2 x3 x4 O( b(k) (3kc)r |v| ) b(k) is a bound for the time to evaluate any subfunction Main idea Decomposition of scores Constant time update
34 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Results: checking the time in the random model • Random model: the number of subfunctions in which a variable appears, c, is not bounded by a constant NKq-landscapes • Random model • N=1,000 to 12,000 • K=1 to 4 • q=2K+1 • r=1 to 4 • 30 instances per conf. K=3 r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 n TimeHsL r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 100000 200000 300000 400000 N Scoresstoredinmemory NKq-landscapes Sanity check Random model Next improvement
35 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) f(2) f(3) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
36 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) x1 x2 f(1) f(2) f(3) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x)

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Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems

  • 1. 1 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems Francisco Chicano, Darrell Whitley, Renato Tinós
  • 2. 2 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i S1( r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i • Considering binary strings of length n and Hamming distance… Solutions in a ball of radius r r=1 r=2 r=3 Ball of radius r Previous work Research Question How many solutions at Hamming distance r? If r << n : Θ (nr)
  • 3. 3 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • We want to find improving moves in a ball of radius r around solution x • What is the computational cost of this exploration? • By complete enumeration: O (nr) if the fitness evaluation is O(1) • Previous work proposed a way to find improving moves in ball of radius r in O(1) (constant time independent of n): Szeider (DO 2011), Whitley and Chen (GECCO 2012), Chen et al. (GECCO 2013), Chicano et al. (GECCO 2014) • The approach was extended to the multi-objective case: Chicano et al. (EvoCOP 2016) • All the previous results are for unconstrained optimization problems Improving moves in a ball of radius r r Ball of radius r Previous work Research Question
  • 4. 4 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Research Question Ball of radius r Previous work Research Question
  • 5. 5 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Definition: • where f(i) only depends on k variables (k-bounded epistasis) • We will also assume that the variables are arguments of at most c subfunctions • Example (m=4, n=4, k=2): • Are Mk Landscapes a small subset of problems? Are they interesting? • Max-kSAT is a k-bounded pseudo-Boolean optimization problem • NK-landscapes is a (K+1)-bounded pseudo-Boolean optimization problem • Any compressible pseudo-Boolean function can be reduced to a quadratic pseudo-Boolean function (e.g., Rosenberg, 1975) Mk Landscape (Whitley, GECCO2015: 927-934) Pseudo-Boolean functions Scores Update Decomposition The family of k-bounded pseudo-Boolean Optimization problems have also been described as an embedded landscape. An embedded landscape [3] with bounded epistasis k is de- fined as a function f(x) that can be written as the sum of m subfunctions, each one depending at most on k input variables. That is: f(x) = mX i=1 f(i) (x), (1) where the subfunctions f(i) depend only on k components of x. Embedded Landscapes generalize NK-landscapes and the MAX-kSAT problem. We will consider in this paper that the number of subfunctions is linear in n, that is m 2 O(n). For NK-landscapes m = n and is a common assumption in MAX-kSAT that m 2 O(n). 3. SCORES IN THE HAMMING BALL For v, x 2 Bn , and a pseudo-Boolean function f : Bn ! R, we denote the Score of x with respect to move v as Sv(x), defined as follows:1 Sv(x) = f(x v) f(x), (2) 1 We omit the function f in Sv(x) to simplify the notation. S(l) v (x) = Equation (5) cl change in the mov f(l) the Score of th this subfunction w On the other hand we only need to c changed variables acterized by the m we can write (3) a S 3.1 Scores De The Score value tion than just the c in that ball. Let us balls of radius r = xj are two variabl ments of any subfu f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x) x1 x2 x3 x4
  • 6. 6 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Based on Mk Landscapes Going Multi-Objective: Vector Mk Landscape Pseudo-Boolean functions Scores Update Decomposition x1 x2 x3 x4 x5 f (1) 1 f (2) 1 f (3) 1 f (4) 1 f (5) 1 f (1) 2 f (2) 2 f (3) 2 (a) Vector Mk Landscape x3x4x5 f1 f2
  • 7. 7 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Going Multi-Objective & Constrained Pseudo-Boolean functions Scores Update Decomposition f1 f2 x1 x2 x3 x4 x5 f (1) 1 f (1) 2 f (2) 1 f (2) 2 f (3) 1 g (1) 1 g (2) 1 g (3) 1g1 Feasible solutions: . BACKGROUND In constrained multi-objective optimization, there is a vec- r function f : Bn ! Rd to optimize, called the objective nction. We will assume, without loss of generality, that l the objectives (components of the vector function) are be maximized. The constraints of the problem will be ven in the form1 g(x) 0, where g : Bn ! Rb is a vec- r function, that will be called constraint function2 . That , a solution is feasible if all the components of the vector nction g are nonnegative when evaluated in that solution. his type of constraints does not represent a limitation. Any her equality or inequality constraint can be expressed in he form gi(x) 0, including those that use strict inequality onstraints (> and <)3 . The set of feasible solutions of a roblem will be denoted by Xg = {x 2 Bn |g(x) 0}. Given a vector function f : Bn ! Rd , we say that solution Objectivefunction Constraint function
  • 8. 8 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Let us represent a potential move of the current solution with a binary vector v having 1s in the positions that should be flipped • The score of move v for solution x is the difference in the fitness value of the neighboring and the current solution • Scores are useful to identify improving moves: if Sv(x) > 0, v is an improving move Scores: definition Current solution, x Neighboring solution, y Move, v 01110101010101001 01111011010101001 00001110000000000 01110101010101001 00110101110101111 01000000100000110 01110101010101001 01000101010101001 00110000000000000 Pseudo-Boolean functions Scores Update Decomposition vector Mk Landscape of Figure 1(a). move in Bn can be characterized by a binary n having 1 in all the bits that change in the so Score of a move, has been previously defined ment in the objective function when that move is finition 4 (Score). For v, x 2 Bn , and a on f : Bn ! Rd , we denote the Score of x with ve v for function f as S (f) v (x), defined as follow S(f) v (x) = f(x v) f(x), is the exclusive OR bitwise operation.
  • 9. 9 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) Scores update r Selected improving move Update the scores Pseudo-Boolean functions Scores Update Decomposition
  • 10. 10 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) • How can we do it less expensive? • We have still O(nr) scores to update! • … thanks to two key facts: • We don’t need all the O(nr) scores to have complete information of the influence of a move in the objective or constraint (vector) function, only O(n) scores • From the ones we need, we only have to update a constant number of them and we can do each update in constant time Key facts for efficient scores update r Pseudo-Boolean functions Scores Update Decomposition
  • 11. 11 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work What is an improving move in MO? • An improving move is one that provides a solution dominating the current one • We define strong and weak improving moves. • We are interested in strong improving moves Taking improving moves Taking feasible moves Algorithm f2 f1 Assuming maximization Can we safely take only strong improving moves?
  • 12. 12 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work We need to take weak improving moves • We could miss some higher order strong improving moves if we don’t take weak improving moves S2 S1 stored stored not stored current solution x1 x2 x3 x4 x5 f2 f2 g (1) 1 g (2) 1 g (3) 1 S (f) v1[v2 (x) = S(f) v1 (x) + S(f) v2 (x) Taking improving moves Taking feasible moves Algorithm
  • 13. 13 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Cycling • We can make the hill climber to cycle if we take weak improving moves S2 S1 stored current solution S2 S1 stored current solution Taking improving moves Taking feasible moves Algorithm
  • 14. 14 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Solution: weights for the scores • Given a weight vector w with wi > 0 S2 S1 w 1st strong improving 2nd w-improving 2nd w-improving A w-improving move is one with w · Sv(x) > 0 Taking improving moves Taking feasible moves Algorithm
  • 15. 15 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility does not only depends on the scores • In order to classify the moves as feasible or unfeasible we have to check all the scores, even if they have not changed • This can be done in O(n), but not in O(1) in general g2 g1 Unfeasible solution current solution v1 v1 v2 Feasible region Taking improving moves Taking feasible moves Algorithm
  • 16. 16 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility in the Hamming Ball x3 x4 x5 f (2) 1 f (2) 2 f (3) 1 g (2) 1 g (3) 1 S (f) v1[v2 (x) = S(f) v1 (x) + S(f) v2 (x) (1) S (g) v1[v2 (x) = S(g) v1 (x) + S(g) v2 (x) (2) g2 g1 Unfeasible stored solution current solution v1 v1 v2 Feasible region v2 Unfeasible stored solution Feasible not stored solution Taking improving moves Taking feasible moves Algorithm
  • 17. 17 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Feasibility in the Hamming Ball g2 g1 current solution v1 v1 v2 Feasible region v2 w* w*-feasible • Bad news: this does not work when the current solution is unfeasible • Worse news: there is no “efficient” way to identify a feasible solution when the current solution is unfeasible Taking improving moves Taking feasible moves Algorithm
  • 18. 18 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Unfeasible regionFeasible region No w-improving or w*-feasible moves ▶ stop strong improving in g w*-improving in g solution y otherwise Hill Climber Taking improving moves Taking feasible moves Algorithm
  • 19. 19 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work MNK Landscape Problem Results Source Code Why NKq and not NK? Floating point precision • An MNK Landscape is a multi-objective pseudo-Boolean problem where each objective is an NK Landscape (Aguirre, Tanaka, CEC 2004: 196-203) • An NK-landscape is a pseudo-Boolean optimization problem with objective function: where each subfunction f(l) depends on variable xl and K other variables • The subfunctions are randomly generated and the values are taken in the range [0,1] • In NKq-landscapes the subfunctions take integer values in the range [0,q-1] • We used shifted NKq-landscapes in the experiments with values in two ranges: • [-49, 50] slightly constrained problems (2% unfeasible) • [-50, 49] highly constrained problems (80% unfeasible) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) f(x) = NX l=1 f(l) (x) 1 • In the adjacent model the variables are consecutive f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
  • 20. 20 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work 0 2000 4000 6000 8000 10000 12000 14000 16000 1 2 3 4 5 6 7 8 9 10 Averagetimepermove(microseconds) N (number of variables in thousands) r=1 r=2 r=3 Runtime: highly constrained MNK Landscapes Neighborhood size: 166 billion Problem Results Source Code
  • 21. 21 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work ries from 1 to 3. These instances are slightly constrained th around 2% of the search space being unfeasible10 . We rformed 30 independent runs of the algorithm for each nfiguration, and the results are the average of these 30 ns. 0 200 400 600 800 1000 1200 1400 1600 10 20 30 40 50 60 70 80 90 100 Averagetimepermove(microseconds) N (number of variables in thousands) r=1, d=1, b=1 r=1, d=2, b=1 r=1, d=1, b=2 r=1, d=2, b=2 r=2, d=1, b=1 r=2, d=2, b=1 r=2, d=1, b=2 r=2, d=2, b=2 r=3, d=1, b=1 r=3, d=2, b=1 r=3, d=1, b=2 r=3, d=2, b=2 gure 3: Average time per move in microsec- Figure 4 for the rithm 1 fo b = 1, N [ 50, 49], In this c the time p per move Figure 3, times sma the unfeas all the mo time. 5.2 Qu Runtime: slightly constrained MNK Landscapes Neighborhood size: 166 trillion Problem Results Source Code from 1 to 3. These instances are slightly constrained round 2% of the search space being unfeasible10 . We med 30 independent runs of the algorithm for each uration, and the results are the average of these 30 0 200 400 600 800 1000 1200 1400 1600 10 20 30 40 50 60 70 80 90 100 N (number of variables in thousands) r=1, d=1, b=1 r=1, d=2, b=1 r=1, d=1, b=2 r=1, d=2, b=2 r=2, d=1, b=1 r=2, d=2, b=1 r=2, d=1, b=2 r=2, d=2, b=2 r=3, d=1, b=1 r=3, d=2, b=1 r=3, d=1, b=2 r=3, d=2, b=2 0 1 Figure 4: A for the Mul rithm 1 for co b = 1, N = 1 [ 50, 49], and In this case, the time per m per move in th Figure 3, even times smaller) the unfeasible all the moves time. 5.2 Qualit
  • 22. 22 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Source Code https://github.com/jfrchicanog/EfficientHillClimbers/tree/constrained-multiobjective Problem Results Source Code
  • 23. 23 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Conclusions and Future Work Conclusions & Future Work • Adding constrains to the MK Landscapes has a cost in terms of efficiency: from O(1) to O(n) • The space required to store the information is still linear in the size of the problem n Conclusions • Generalize to other search spaces • Combine with high-level algorithms (MOEA/D) Future Work
  • 24. 24 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Acknowledgements Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems
  • 25. 25 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Examples: 1 and 4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) f(1) f(2) f(3) f(4) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) Pseudo-Boolean functions Scores Update Decomposition
  • 26. 26 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Example: 1,4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 r Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) n i S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1(x) = f(x 1) f(x) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) Ball Pr i=1 n i S1(x) = Sv(x) = f(x v) f(x) = S4(x) = S1(x) = f(x 1) f(x) x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We don’t need to store S1,4(x) since can be computed from others If none of 1 and 4 are improving moves, 1,4 will not be an improving move Pseudo-Boolean functions Scores Update Decomposition
  • 27. 27 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Quality • 50% Empirical Attainment Functions (Knowles, ISDA 2005: 552-557) Problem Results Source Code 0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06 10 20 30 40 50 60 70 80 90 100 Averagequalityofbestfoundsolution N (number of variables in thousands) r=1, b=1 r=1, b=2 r=2, b=1 r=2, b=2 r=3, b=1 r=3, b=2 Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 (a) N = 10, 000, b = 1 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (b) N = 50, 000, b = 2 Figure 6: 50%-empirical attainment surfaces of the 30 runs of the Multi-Start Hill Climber based on Acknowledgements This research was partially funded by gram, the Spanish Ministry of Education, (CAS12/00274), the Spanish Ministry of E petitiveness and FEDER (TIN2014-573 sity of M´alaga, Andaluc´ıa Tech, the A Scientific Research, Air Force Materiel (FA9550-11-1-0088), the FAPESP (2015/ 7. REFERENCES [1] Hernan E. Aguirre and Kiyoshi Tan properties of multiobjective MNK-la Proceedings of CEC, volume 1, page [2] Wenxiang Chen, Darrell Whitley, D Adele Howe. Second order partial de NK-landscapes. In Proceeding of GE 503–510, New York, NY, USA, 2013 [3] Francisco Chicano, Darrell Whitley, Sutton. E cient identification of im ball for pseudo-boolean problems. In GECCO, pages 437–444. ACM, 201 [4] Francisco Chicano, Darrell Whitley, Tin´os. E cient hill climber for mult pseudo-boolean optimization. In Pro EvoCOP, pages 88–103, 2016. [5] Yves Crama, Pierre Hansen, and Br The basic algorithm for pseudo-boo revisited. Discrete Applied Mathema 29(2-3):171–185, 1990. [6] Brian W. Goldman and William F. optimization using the parameter-le pyramid. In Proceedings of GECCO 0 500000 1e+06 10 20 30 40 50 60 70 80 90 100 Averagequ N (number of variables in thousands) Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (FA9550- 7. RE [1] Hern prop Proc [2] Wen Ade NK- 503– [3] Fran Sutt ball GEC [4] Fran Tin´o pseu Evo [5] Yve The revi 0 500000 1e+06 10 20 30 40 50 60 70 80 90 100 Averagequ N (number of variables in thousands) Figure 5: Average (over 30 runs) solution quality of the best solution found by the Multi-Start Hill Climber based on Algorithm 1 for a MNK Land- scape with d = 1, b = 1, 2, subfunctions codomain [ 49, 50], N = 10, 000 to 100, 000, and r = 1 to 3. 0 50000 100000 150000 200000 250000 300000 0 50000 100000 150000 200000 250000 300000 f2 f1 r=1 r=2 r=3 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 f2 f1 r=1 r=2 r=3 (F 7. [ [ [ [ [ Single-objective constrained problem Bi-objective constrainedproblems 1 constraint 2 constraints
  • 28. 28 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Example: 1,2 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) f(1) f(2) f(3) x1 x2 x3 x4 f(1) f(2) f(3) x1 x2 x3 x4 f(1) x1 x2 Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) x1 and x2 “interact” Pseudo-Boolean functions Scores Update Decomposition
  • 29. 29 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Decomposition rule for scores • When can we decompose a score as the sum of lower order scores? • … when the variables in the move can be partitioned in subsets of variables that DON’T interact • Let us define the Co-occurrence Graph f(1) f(2) f(3) f(4) x1 x2 x3 x4 There is an edge between two variables if there exists a function that depends on both variables (they “interact”) x4 x3 x1 x2 Pseudo-Boolean functions Scores Update Decomposition
  • 30. 30 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work • Whitley and Chen proposed an O(1) approximated steepest descent for MAX-kSAT and NK-landscapes based on Walsh decomposition • For k-bounded pseudo-Boolean functions its complexity is O(k2 2k) • Chen, Whitley, Hains and Howe reduced the time required to identify improving moves to O(k3) using partial derivatives • Szeider proved that the exploration of a ball of radius r in MAX-kSAT and kSAT can be done in O(n) if each variable appears in a bounded number of clauses Previous work Ball of radius r Improving moves Previous work Research Question D. Whitley and W. Chen. Constant time steepest descent local search with lookahead for NK-landscapes and MAX-kSAT. GECCO 2012: 1357–1364 W. Chen, D. Whitley, D. Hains, and A. Howe. Second order partial derivatives for NK-landscapes. GECCO 2013: 503–510 S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139–145, 2011
  • 31. 31 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to store • In terms of the VIG a score can be decomposed if the subgraph containing the variables in the move is NOT connected • The number of these scores (up to radius r) is O((3kc)r n) • Details of the proof in the paper • With a linear amount of information we can explore a ball of radius r containing O(nr) solutions x4 x3 x1 x2 x4 x3 x1 x2 S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f S1,2(x) 6= S1(x) + S2(x) l=1 l=1 S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We need to store the scores of moves whose variables form a connected subgraph of the VIG Pseudo-Boolean functions Scores Update Decomposition
  • 32. 32 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to update • Let us assume that x4 is flipped • Which scores do we need to update? • Those that need to evaluate f(3) and f(4) f(1) f(2) f(3) f(4) x1 x2 x3 x4 x4 x3 x1 x2 • The scores of moves containing variables adjacent or equal to x4 in the VIG Main idea Decomposition of scores Constant time update
  • 33. 33 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores to update and time required • The number of neighbors of a variable in the VIG is bounded by c k • The number of stored scores in which a variable appears is the number of spanning trees of size less than or equal to r with the variable at the root • This number is constant • The update of each score implies evaluating a constant number of functions that depend on at most k variables (constant), so it requires constant time x4 x3 x1 x2 f(1) f(2) f(3) f(4) x1 x2 x3 x4 O( b(k) (3kc)r |v| ) b(k) is a bound for the time to evaluate any subfunction Main idea Decomposition of scores Constant time update
  • 34. 34 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Results: checking the time in the random model • Random model: the number of subfunctions in which a variable appears, c, is not bounded by a constant NKq-landscapes • Random model • N=1,000 to 12,000 • K=1 to 4 • q=2K+1 • r=1 to 4 • 30 instances per conf. K=3 r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 n TimeHsL r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 100000 200000 300000 400000 N Scoresstoredinmemory NKq-landscapes Sanity check Random model Next improvement
  • 35. 35 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) f(2) f(3) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
  • 36. 36 / 24GECCO 2016, Denver, CO, USA, 20-24 July 2016 Genetic and Evolutionary Computation Conference 2016 Conference Program Denver, CO, USA July 20-24, 2016 Introduction Background Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) x1 x2 f(1) f(2) f(3) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x)