Stochastic Optimization
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The document discusses stochastic optimization, focusing on mathematical programming problems where uncertainty in data affects decision-making. It outlines various models, including probabilistically constrained models and recourse models, and their applications in different fields such as supply chain management and financial planning. Key concepts include the distinction between deterministic and stochastic optimization, as well as methods for achieving reliability in constraints.
![ The random variables 𝝃 𝟏 ،𝝃 𝟐 and 𝜼 𝟐 are unbounded. Therefore, we obtain a reliability level (95%) for them:
𝝃 𝟏 𝜖 −30.94 , 30.94 𝜼 𝟏 𝜖 [−23.18 , 23.18]
𝝃 𝟐 𝜖 −0.8 , 0.8 𝜼 𝟐 𝜖 [0.1.84]
Define the random vector 𝒘 = (𝝃 𝟏, 𝝃 𝟐, 𝜼 𝟏, 𝜼 𝟐) and rewrite the problem:
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝑥1 + 𝑥2 ≤ 100
𝛼 𝒘 𝑥1 + 6𝑥2 ≥ ℎ1 𝒘
3𝑥1 + 𝛽 𝒘 𝑥2 ≥ ℎ2 𝒘
𝑥1 ≥ 0 , 𝑥2 ≥ 0
Before determining the random parameters of the problem (a certain realization of 𝒘), the optimal value of
the problem can not be calculated !!!
10/31](https://image.slidesharecdn.com/presentationautosaved-180721144316/85/Stochastic-Optimization-10-320.jpg)
![3. Probabilistically Constrained Models
Definition:
If instead of persisting that the constraints of the problem hold for all values of random variables, suppose
that the constraints hold with a certain reliability level. This case is called an optimization problem with
Probabilistic constraints.
𝑃{
𝑗=1
𝑛
𝑎𝑖𝑗 𝑤 𝑥𝑗 ≥ 𝑏𝑖 𝑤 , 𝑖 = 1,2, … , 𝑚 ≥ 𝛼 𝑎𝑛𝑑 𝛼𝜖[0,1]
𝑃{
𝑗=1
𝑛
𝑎𝑖𝑗 𝑤 𝑥𝑗 ≥ 𝑏𝑖 𝑤 ≥ 𝛼𝑖 , 𝑖 = 1,2, … , 𝑚 𝑎𝑛𝑑 𝛼𝜖[0,1]
1. Dsjoint Probabilistic
Constraints
2. Joint Probabilistic
Constraints
11/31](https://image.slidesharecdn.com/presentationautosaved-180721144316/85/Stochastic-Optimization-11-320.jpg)
![cvx_begin
variables x(3) y
maximize((5*x(1)+6*x(2)+3*x(3)))
subject to
x(1)+3*x(2)+9*x(3)+1.645*norm([5*x(1) 4*x(2) 2*x(3)]) <= 8;
5*x(1)+x(2)+6*x(3) <= 10.855;
log(1-exp(-(4*x(1)+2*x(2)-3*x(3))))+log(1-exp(-(2*x(1)+3*x(2)+x(3))))>= log(0.85);
x >= 0;
cvx_end
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 + 6𝑥2 + 3𝑥3
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝑥1 + 3𝑥2 + 9𝑥3 + 1.645 25𝑥1
2
+ 16𝑥2
2
+ 4𝑥3
2
≤ 8
5𝑥1 + 𝑥2 + 6𝑥3 ≤ 10.855
ln 1 − 𝑒−(4𝑥1+2𝑥2−3𝑥3)
+ ln 1 − 𝑒−(2𝑥1+3𝑥2+𝑥3)
≥ ln(0.85)
𝑥𝑗 ≥ 0 𝑗 = 1,2,3
𝑥∗
=
0.4625
0.6327
1.976 × 10−9
𝑎𝑛𝑑 𝑃∗
= 6.1087
25𝑥1
2
+ 16𝑥2
2
+ 4𝑥3
2
≡ 5𝑥1 4𝑥2 2𝑥3 2
23/31](https://image.slidesharecdn.com/presentationautosaved-180721144316/85/Stochastic-Optimization-23-320.jpg)
![ Two-Stage Program with Fixed Recourse
The classical two-stage stochastic linear program with fixed recourse (originated by Dantzig [1955] and
Beale [1955]) is the problem of finding:
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑐 𝑇
𝑥 + 𝐸𝜉 {min
𝑦
𝑞( 𝒘) 𝑇
𝑦(𝒘)}
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝐴𝑥 = 𝑏
𝑇( 𝒘) 𝑥 + 𝑊𝑦( 𝒘) = ℎ(𝒘)
𝑥 ≥ 0 , 𝑦( 𝒘) ≥ 0
Where:
• 𝑥: The first-stage decisions )𝑛1 × 1)
• 𝑦: The second-stage decisions )𝑛2 × 1)
• 𝑊: Recourse Matrix (𝑚2 × 𝑛2)
• 𝜉 𝑇 𝒘 = (𝑞 𝑇 𝒘 , 𝑞 𝑇 𝒘 , 𝑞 𝑇 𝒘 , 𝑇1 𝒘 , … , 𝑇 𝑚2
𝒘
Each component of q , T , and h is thus a possible random variable.
For a given realization ω , the second-stage problem data q)ω) , h)ω) and T)ω) become known.
25/31](https://image.slidesharecdn.com/presentationautosaved-180721144316/85/Stochastic-Optimization-25-320.jpg)
Stochastic Optimization
- 1. IN THE NAME OF GOD Isfahan University of Technology Department Of Electrical and Computer Engineering Title: Stochastic Optimization Course instructor: DR. Naghmeh Sadat Moayedian By: Mohammad Reza Jabbari July 2018
- 2. CONTENTS 1. Introduction 2. Application 3. Probabilistically Constrained Models 4. Recourse Models 5. Conclusion 6. End 2/31 Contact Info: Mo.re.Jabbari@gmail.com
- 3. 1.Introduction 3/31 Definition Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seek to maximize or minimize some objective which is a function of the decisions. The feasible decisions are constrained by limits in resources, minimum requirements, etc. Objectives and constraints are functions of the variables and problem data, such as costs, production rates, sales, or capacities.
- 4. Optimization Problems Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints is Uncertain. Stochastic Optimization Deterministic Optimization Deterministic Optimization Problems are formulated with known parameters. Real world problems almost invariably include some unknown and uncertain parameters. Robust Optimizatio n 4/31
- 5. When some of the data is random, then Optimal Solutions and the Optimal Value to the optimization problem are themselves random. A distribution of optimal decisions is generally unimplementanble. Probabilistic Distribution Ideally, we would like one decision and one optimal objective value. What do you tell your boss? 5/31
- 6. Problem Solving Methods 1. Probabilistically Constrained Models 2. Recourse Models Try to find a decision which ensures that a set of constraints will hold with a certain probability. One logical way to pose the problem is to require that we make one decision now and minimize the expected costs (or utilities) of the consequences of that decision disjoint Probabilistic Constraint Joint Probabilistic Constraint Two-Stage Multi-Stage 6/31
- 7. The purpose of both methods is to: Stochastic Optimization Problem Deterministic Optimization Problem Deterministic Equivalent Problem So DEP can be solved using linear or nonlinear optimization methods. 7/31
- 8. Sustainability and Power Planning Supply Chain Management Network optimization Logistics Financial Management Location Analysis etc. 2.Applications 8/31
- 9. Suppose that a factory can simultaneously produce two products from two raw materials 𝑥1and 𝑥2 with below constraints: 1. The production cost of each unit of the first and second product is 𝑐1 = 2 and 𝑐2 = 3, respectively 2. The maximum storage capacity for storing raw materials is equal to b = 100. 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑥2 ≤ 100 2𝑥1 + 6𝑥2 ≥ 180 3𝑥1 + 3𝑥2 ≥ 162 𝑥1 ≥ 0 , 𝑥2 ≥ 0 𝑥1 ∗ = 36 𝑥2 ∗ = 18 𝑝∗ = 126 Example1: !! Note that the assumption that parameters such as the minimum demand or cost per unit of raw material, etc. are not realistic. Therefore, each of these parameters and coefficients can be a random variable: ℎ1 = 180 + 𝝃 𝟏 𝑎21 = 2 + 𝜼 𝟏 ℎ1 = 162 + 𝝃 𝟐 𝑎32 = 3 + 𝜼 𝟐 Where 𝝃 𝟏~ 𝑁 0,12 𝜼 𝟏~ 𝑈(−0.8 , 0.8) 𝝃 𝟐~ 𝑁 0,9 𝜼 𝟐~ exponential (λ = 2.5) 9/31
- 10. The random variables 𝝃 𝟏 ،𝝃 𝟐 and 𝜼 𝟐 are unbounded. Therefore, we obtain a reliability level (95%) for them: 𝝃 𝟏 𝜖 −30.94 , 30.94 𝜼 𝟏 𝜖 [−23.18 , 23.18] 𝝃 𝟐 𝜖 −0.8 , 0.8 𝜼 𝟐 𝜖 [0.1.84] Define the random vector 𝒘 = (𝝃 𝟏, 𝝃 𝟐, 𝜼 𝟏, 𝜼 𝟐) and rewrite the problem: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑥2 ≤ 100 𝛼 𝒘 𝑥1 + 6𝑥2 ≥ ℎ1 𝒘 3𝑥1 + 𝛽 𝒘 𝑥2 ≥ ℎ2 𝒘 𝑥1 ≥ 0 , 𝑥2 ≥ 0 Before determining the random parameters of the problem (a certain realization of 𝒘), the optimal value of the problem can not be calculated !!! 10/31
- 11. 3. Probabilistically Constrained Models Definition: If instead of persisting that the constraints of the problem hold for all values of random variables, suppose that the constraints hold with a certain reliability level. This case is called an optimization problem with Probabilistic constraints. 𝑃{ 𝑗=1 𝑛 𝑎𝑖𝑗 𝑤 𝑥𝑗 ≥ 𝑏𝑖 𝑤 , 𝑖 = 1,2, … , 𝑚 ≥ 𝛼 𝑎𝑛𝑑 𝛼𝜖[0,1] 𝑃{ 𝑗=1 𝑛 𝑎𝑖𝑗 𝑤 𝑥𝑗 ≥ 𝑏𝑖 𝑤 ≥ 𝛼𝑖 , 𝑖 = 1,2, … , 𝑚 𝑎𝑛𝑑 𝛼𝜖[0,1] 1. Dsjoint Probabilistic Constraints 2. Joint Probabilistic Constraints 11/31
- 12. 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑥2 ≤ 100 𝑃{ 𝒘 𝛼( 𝑤) 𝑥1 + 6𝑥2 ≥ ℎ1( 𝑤) , 3𝑥1 + 𝛽( 𝑤) 𝑥2 ≥ ℎ2( 𝑤) } ≥ 0.95 𝑥1 ≥ 0 , 𝑥2 ≥ 0 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑥2 ≤ 100 𝛼( 𝒘) 𝑥1 + 6𝑥2 ≥ ℎ1( 𝒘) 3𝑥1 + 𝛽( 𝒘) 𝑥2 ≥ ℎ2( 𝒘) 𝑥1 ≥ 0 , 𝑥2 ≥ 0 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 2𝑥1 + 3𝑥2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑥2 ≤ 100 𝑃{ 𝑤 𝛼( 𝒘) 𝑥1 + 6𝑥2 ≥ ℎ1( 𝒘) } ≥ 1 − 𝛼1 𝑃{ 𝑤 3𝑥1 + 𝛽( 𝒘) 𝑥2 ≥ ℎ2( 𝒘)} ≥ 1 − 𝛼2 𝑥1 ≥ 0 , 𝑥2 ≥ 0 12/31
- 13. Joint Probabilistic Constraints In optimization problems with Joint Probabilistic Constraints, often apply conditions to probability distribution in order to obtain DEP. The purpose of these conditions is to satisfy convexity of feasible set or the objective function. logarithmically concave Concept Suppose that 𝑎𝑖𝑗 ’s coefficients are definite and deterministic and the parameters 𝑏𝑖(𝑖 = 1, … , 𝑚1) are random, independent with the probability corresponding to the logarithmically concave 𝑃𝑖 and the corresponding probability distribution function 𝐹𝑖: 𝑃 𝐴𝑥 ≥ 𝑏 ≥ 𝛼 𝑖=1 𝑚1 𝑃𝑖 𝐴𝑖 𝑥 ≥ 𝑏𝑖 ≥ 𝛼 𝑖=1 𝑚1 𝐹𝑖 𝐴𝑖 𝑥 ≥ 𝛼 𝑖=1 𝑚1 𝐿𝑛( 𝐹𝑖 𝐴𝑖 𝑥 ) ≥ 𝐿𝑛(𝛼) To show that this equivalent equation is convex we just need to show that the probability distribution function 𝐹𝑖 ،is logarithmically concave. 13/31
- 14. Disjoint Probabilistic Constraints 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑗=1 𝑛 𝑐𝑗 𝑥𝑗 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑃 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖 ≥ 1 − 𝛼𝑖 𝑖 = 1, … , 𝑚 𝛼𝑖 𝜖 0,1 , 𝑥𝑗 ≥ 0 𝑖 = 1, … , 𝑚 𝑗 = 1, … , 𝑛 In fact, the i-th constraint is satisfied with the least probability 1 − 𝛼𝑖. In order to transform the above problem into a definite equivalent problem, seven state arise, depending on which of the parameters of the problem is random. By examining its four independent states, the remaining states can be obtained from the combination of previous states 14/31
- 15. State1: • Assume that only 𝑐𝑗’s are random variables with mean 𝐸{𝑐𝑗} 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑗=1 𝑛 𝐸{𝑐𝑗}𝑥𝑗 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖 𝑖 = 1, … , 𝑚 𝑥𝑗 ≥ 0 𝑗 = 1, … , 𝑛 • In this case, it is enough to substitute 𝑐𝑗’s with their means. 15/31
- 16. State2: • Assume that 𝑎𝑖𝑗’s are correlated variable with the mean 𝑬{𝒂𝒊𝒋}, the variance 𝑽𝒂𝒓{𝒂𝒊𝒋}, and the covariance matrix 𝑪𝒐𝒗(𝒂𝒊𝒋, 𝒂𝒊′ 𝒋′). • Also, we assume i-th constraint is in the following form: 𝑖𝑓 𝑑𝑒𝑓𝑖𝑛𝑒 𝑇𝑖 = 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑗=1 𝑛 𝑐𝑗 𝑥𝑗 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑃 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖 ≥ 1 − 𝛼𝑖 𝑖 = 1, … , 𝑚 𝛼𝑖 𝜖 0,1 , 𝑥𝑗 ≥ 0 𝑖 = 1, … , 𝑚 𝑗 = 1, … , 𝑛 𝐸 𝑇𝑖 = 𝑗=1 𝑛 𝐸 𝑎𝑖𝑗 𝑥𝑗 𝑖 = 1, … , 𝑚 𝑉𝑎𝑟 𝑇𝑖 = 𝑚=1 𝑛 𝑉𝑎𝑟 𝑎𝑖𝑚 𝑥 𝑚 2 + 𝑚=1 𝑛 𝑘=1 𝑘≠𝑚 𝑛 𝐶𝑜𝑣(𝑎𝑖𝑚, 𝑎𝑖𝑘) 𝑥 𝑚 𝑥 𝑘 16/31
- 17. • Base on Central Limit theory, 𝑇𝑖 is approximately Gaussian Distribution. 𝑃 𝑇𝑖 ≤ 𝑏𝑖 = 1 − 𝑄 𝑏𝑖 − 𝐸 𝑇𝑖 𝑉𝑎𝑟 𝑇𝑖 𝑖 = 1, … , 𝑚 i𝑓 𝑑𝑒𝑓𝑖𝑛𝑒 𝑄 𝐾 𝛼 𝑖 = 𝛼𝑖 𝑏𝑖 − 𝐸 𝑇𝑖 𝑉𝑎𝑟 𝑇𝑖 ≥ 𝐾 𝛼 𝑖 𝑖 = 1, … , 𝑚 𝐸 𝑇𝑖 + 𝐾 𝛼 𝑖 𝑉𝑎𝑟 𝑇𝑖 ≤ 𝑏𝑖 If 𝑎𝑖𝑗’s were independent variables so 𝐶𝑜𝑣 𝑎𝑖𝑗, 𝑎𝑖′ 𝑗′ = 0 and: 𝑗=1 𝑛 𝐸 𝑎𝑖𝑗 𝑥𝑗 + 𝐾 𝛼 𝑖 𝑗=1 𝑛 𝑉𝑎𝑟 𝑎𝑖𝑗 𝑥𝑗 2 ≤ 𝑏𝑖 𝑖 = 1, … , 𝑚 • Therefore, the below deterministic equivalent constraint be replaced with probabilistic constraint: 17/31
- 18. State3: • Assume that 𝑏𝑖’s are Gaussian variable with the mean 𝑬{𝒃𝒊}, the variance 𝑽𝒂𝒓 𝒃𝒊 : • Also, we assume i-th constraint is in the following form: 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑗=1 𝑛 𝑐𝑗 𝑥𝑗 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑃 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝑏𝑖 ≥ 𝛼𝑖 𝑖 = 1, … , 𝑚 𝛼𝑖 𝜖 0,1 , 𝑥𝑗 ≥ 0 𝑖 = 1, … , 𝑚 𝑗 = 1, … , 𝑛 • Like Previous state: 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 − 𝐸 𝑏𝑖 𝑉𝑎𝑟 𝑏𝑖 ≤ 𝐾 𝛼 𝑖 𝑖 = 1, … , 𝑚 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 ≤ 𝐸 𝑏𝑖 + 𝐾 𝛼 𝑖 𝑉𝑎𝑟 𝑏𝑖 𝑖 = 1, … , 𝑚 18/31
- 19. State4: • In this State, assume that 𝑏𝑖’s and 𝑎𝑖𝑗’s are Random Variable with the mean and variance 𝑬{𝒃𝒊}, 𝑽𝒂𝒓 𝒃𝒊 , 𝑬{𝒂𝒊𝒋} and 𝑽𝒂𝒓{𝒂𝒊𝒋} • Also, we assume i-th constraint is in the following form: 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑗=1 𝑛 𝑐𝑗 𝑥𝑗 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑃 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 − 𝑏𝑖 ≤ 0 ≥ 1 − 𝛼𝑖 𝑖 = 1, … , 𝑚 𝛼𝑖 𝜖 0,1 , 𝑥𝑗 ≥ 0 𝑖 = 1, … , 𝑚 𝑗 = 1, … , 𝑛 if define ℎ𝑖 ≜ 𝑗=1 𝑛 𝑎𝑖𝑗 𝑥𝑗 − 𝑏𝑖 𝐸 ℎ𝑖 = 𝑗=1 𝑛 𝐸 𝑎𝑖𝑗 𝑥𝑗 − 𝐸 𝑏𝑖 𝑖 = 1, … , 𝑚 𝑉𝑎𝑟 ℎ𝑖 = 𝑥 𝑇 𝐷𝑖 𝑥 , 𝑖 = 1, … , 𝑚 𝐷𝑖 = )𝑉𝑎𝑟(𝑎𝑖1 )𝐶𝑜𝑣(𝑎𝑖1, 𝑎𝑖2 ⋯ )𝐶𝑜𝑣(𝑎𝑖1, 𝑎𝑖𝑛 )𝐶𝑜𝑣(𝑎𝑖1, 𝑏𝑖 )𝐶𝑜𝑣(𝑎𝑖2, 𝑎𝑖1 )𝑉𝑎𝑟(𝑎𝑖2 ⋯ )𝐶𝑜𝑣(𝑎𝑖2, 𝑎𝑖𝑛 )𝐶𝑜𝑣(𝑎𝑖1, 𝑏𝑖 ⋮ ⋮ ⋱ ⋮ ⋮ )𝐶𝑜𝑣(𝑎𝑖𝑛, 𝑎𝑖1 )𝐶𝑜𝑣(𝑎𝑖𝑛, 𝑎𝑖2 ⋯ )𝑉𝑎𝑟(𝑎𝑖𝑛 )𝐶𝑜𝑣(𝑎𝑖𝑛, 𝑏𝑖 )𝐶𝑜𝑣(𝑏𝑖, 𝑎𝑖1 )𝐶𝑜𝑣(𝑏𝑖, 𝑎𝑖1 ⋯ )𝐶𝑜𝑣(𝑎𝑖𝑛, 𝑎𝑖1 )𝑉𝑎𝑟(𝑏𝑖
- 20. • Base on Central Limit theory,ℎ𝑖 is approximately Gaussian Distribution. 𝑃 ℎ𝑖 ≤ 0 = 1 − 𝑄 −𝐸 ℎ𝑖 𝑉𝑎𝑟 ℎ𝑖 ≥ 1 − 𝛼𝑖 −𝐸 ℎ𝑖 𝑉𝑎𝑟 ℎ𝑖 ≥ 𝐾 𝛼 𝑖 𝐸 ℎ𝑖 + 𝐾 𝛼 𝑖 𝑉𝑎𝑟 ℎ𝑖 ≤ 0 • Therefore, the deterministic equivalent constraint is: The next three States are combination the four previous sates and can easily obtained: • State5: 𝑐𝑗’s and 𝑎𝑖𝑗’s are random variable • State6: 𝑐𝑗’s and 𝑏𝑖’s are random variable • State7: 𝑐𝑗’s and 𝑎𝑖𝑗’s and 𝑏𝑖’s are random variable 20/31
- 21. Example2: Consider the following optimization problem: 𝑎11~ 𝑁(1 , 25) 𝑎12~ 𝑁(3 , 16) 𝑎13~ 𝑁(9 , 4) 𝑏2~ 𝑁(7 , 9) 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 + 6𝑥2 + 3𝑥3 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑃{ 𝑎11 𝑥1 + 𝑎12 𝑥2 + 𝑎13 𝑥3 ≤ 8} ≥ 0.95 𝑃{ 5𝑥1 + 𝑥2 + 6𝑥3 ≤ 𝑏2} ≥ 0.1 𝑃{ 4𝑥1 + 2𝑥2 − 3𝑥3 ≥ 𝑏3 , 2𝑥1 + 3𝑥2 + 𝑥3 ≥ 𝑏4} ≥ 0.85 𝑥𝑗 ≥ 0 𝑗 = 1, … , 𝑚 𝑏3 𝑎𝑛𝑑 𝑏4~ 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 (𝜆 = 1) • 1st constraint ≡ State2 𝑥1 + 3𝑥2 + 9𝑥3 + 1.645 25𝑥1 2 + 16𝑥2 2 + 4𝑥3 2 ≤ 8𝑄 𝐾0.05 = 0.05 → 𝐾0.05 = 𝑄−1 0.05 = 1.645 21/31
- 22. • 2nd constraint ≡ State3 𝑄 𝐾0.1 = 0.1 → 𝐾0.1 = 𝑄−1 0.1 = 1.285 5𝑥1 + 𝑥2 + 6𝑥3 ≤ 7 + 1.285 9 • 3rd constraint ≡ joint Probabilistic constraint 𝐹𝑏 𝐴𝑖 𝑥 = 0 𝐴 𝑖 𝑥 𝑒−𝑡 𝑑𝑡 = 1 − 𝑒−𝐴 𝑖 𝑥 ln 1 − 𝑒− 4𝑥1+2𝑥2−3𝑥3 + ln 1 − 𝑒− 2𝑥1+3𝑥2+𝑥3 ≥ l n( 0.85 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 + 6𝑥2 + 3𝑥3 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 3𝑥2 + 9𝑥3 + 1.645 25𝑥1 2 + 16𝑥2 2 + 4𝑥3 2 ≤ 8 5𝑥1 + 𝑥2 + 6𝑥3 ≤ 10.855 ln 1 − 𝑒−(4𝑥1+2𝑥2−3𝑥3) + ln 1 − 𝑒−(2𝑥1+3𝑥2+𝑥3) ≥ ln(0.85) 𝑥𝑗 ≥ 0 𝑗 = 1,2,3 So DEP is: 22/31
- 23. cvx_begin variables x(3) y maximize((5*x(1)+6*x(2)+3*x(3))) subject to x(1)+3*x(2)+9*x(3)+1.645*norm([5*x(1) 4*x(2) 2*x(3)]) <= 8; 5*x(1)+x(2)+6*x(3) <= 10.855; log(1-exp(-(4*x(1)+2*x(2)-3*x(3))))+log(1-exp(-(2*x(1)+3*x(2)+x(3))))>= log(0.85); x >= 0; cvx_end 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 + 6𝑥2 + 3𝑥3 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 3𝑥2 + 9𝑥3 + 1.645 25𝑥1 2 + 16𝑥2 2 + 4𝑥3 2 ≤ 8 5𝑥1 + 𝑥2 + 6𝑥3 ≤ 10.855 ln 1 − 𝑒−(4𝑥1+2𝑥2−3𝑥3) + ln 1 − 𝑒−(2𝑥1+3𝑥2+𝑥3) ≥ ln(0.85) 𝑥𝑗 ≥ 0 𝑗 = 1,2,3 𝑥∗ = 0.4625 0.6327 1.976 × 10−9 𝑎𝑛𝑑 𝑃∗ = 6.1087 25𝑥1 2 + 16𝑥2 2 + 4𝑥3 2 ≡ 5𝑥1 4𝑥2 2𝑥3 2 23/31
- 24. 4. Recourse Models Recourse Models are those in which some decisions or recourse actions can be taken after uncertainty is disclosed. Definition: Stochastic Optimization with Recource Anticipative Variables Adoptive Variables The set of decisions is then divided into two groups: • A number of decisions have to be taken before the experiment. All these decisions are called first-stage decisions and the period when these decisions aretaken is called the first stage. • A number of decisions can be taken after the experiment. They are called second-stage decisions. The corresponding period is called the second stage. 24/31
- 25. Two-Stage Program with Fixed Recourse The classical two-stage stochastic linear program with fixed recourse (originated by Dantzig [1955] and Beale [1955]) is the problem of finding: 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑐 𝑇 𝑥 + 𝐸𝜉 {min 𝑦 𝑞( 𝒘) 𝑇 𝑦(𝒘)} 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑥 = 𝑏 𝑇( 𝒘) 𝑥 + 𝑊𝑦( 𝒘) = ℎ(𝒘) 𝑥 ≥ 0 , 𝑦( 𝒘) ≥ 0 Where: • 𝑥: The first-stage decisions )𝑛1 × 1) • 𝑦: The second-stage decisions )𝑛2 × 1) • 𝑊: Recourse Matrix (𝑚2 × 𝑛2) • 𝜉 𝑇 𝒘 = (𝑞 𝑇 𝒘 , 𝑞 𝑇 𝒘 , 𝑞 𝑇 𝒘 , 𝑇1 𝒘 , … , 𝑇 𝑚2 𝒘 Each component of q , T , and h is thus a possible random variable. For a given realization ω , the second-stage problem data q)ω) , h)ω) and T)ω) become known. 25/31
- 26. Example3: Suppose we have to decide on the number of production X products. • The production of each unit X costs $ 2. • Customers' demand is random and there is a discrete distribution Ds with the probability ps (s = 1, ..., S). • Customer demand must be met. We will have the opportunity to buy X from an external supplier to meet the real demand of our customers. This purchase costs $ 3. Question: How much should we produce at the moment, while we do not know the future demand of our customers? We assume S=2 and D1=500, p1=0.6 D2=700, p2=0.4 First Stage Second Stage D1=500, p1=0.6 D2=700, p2=0.4 26/31
- 27. Misleading solution: The optimal solution is the expected amount of demand. Correct solution: The optimal solution is obtained through stochastic Optimization. 26/31 27 The number of units of the X that are currently produced (the first stage). x1 :the 1st stage The number of units of X, which in the second stage is purchased with the random fulfillment of demand scenarios, Ds (s = 1, ..., S). y2s :the 2nd stage
- 28. cvx_begin variables x y(2) minimize(2*x+1.8*y(1)+1.2*y(2)) subject to x+y(1)>=500 x+y(2)>=700 x>=0; y>=0 cvx_end 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 2𝑥1 + 𝑝𝑠(3𝑦2𝑠) 2 𝑠=1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 + 𝑦2𝑠 ≥ 𝐷𝑠 𝑠 = 1,2 𝑥1 ≥ 0 𝑦2𝑠 ≥ 0 𝑠 = 1,2 𝑥∗ = 500 𝑦 = 1.22𝑒 − 8 200 Optimal cost= 1240 28/31
- 29. If we choose expected value of demand: 𝑥∗ = 580 if D=700 → Optimal cost = 1350 29/31
- 30. Stochastic programming problems are formulated as mathematical programming tasks with the objective and constraints defined as expectations of some random functions or probabilities of some sets of scenarios. Expectations are defined by multivariate integrals (scenarios distributed continuously) or finite series (scenarios distributed discretely). 5. Conclusion 30/31
- 31. END 31