Two-stage Stochastic Optimization

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1 Two-stage Stochastic Optimization September 13, 2011 Most of the material presented in this class is directly taken from Chapter 2 of the book Lectures on Stochastic Programg: Modeling and Theory by Shapiro, Dentcheva and Ruszczynski. Basic knowledge of linear programg, duality, polyhedral sets and polyhedral functions is required to understand this material. 1 An Example: Multi-product Assembly This example has been taken from Section of the book. Consider a manufacturer which produces n products. To manufacture these products, it orders m different types of parts (or subassemblies) from external suppliers. Various known parameters are mentioned below. a ij = number of units of part-j required for a unit of product-i (i = 1,, n, j = 1,, m) c j = cost of purchase of unit part-j (j = 1,, m) l i = cost of production of unit product-i (i = 1,, n) q i = selling price of unit product-i (i = 1,, n) s j = salvage value of unit part-j (j = 1,, m) The main issue here is that the order for the parts is to be placed before the random demand D = (D 1,, D n ) for the products is known. After the demand is observed, the manufacturer may decide which portion of the demand is to be satisfied. The unsatisfied demand is lost. The leftover parts are assessed at a salvage value. Suppose the numbers of parts ordered are equal to x j (j = 1,, m). Then, for an observed value d = (d 1,, d n ) of the demand D, the best production plan can be found by solving the following linear programg (LP) problem. s.t. (l i q i )z i s j y j i=1 y j = x j j=1 a ij z i, i=1 j = 1,, m 0 z i d i, i = 1,, n y j 0, j = 1,, m where z i (i = 1,, n) is the number of units of product-i manufactured and y j (j = 1,, m) is the number of units of part-j left over after production. The above LP formulation can be written in vector form as: 1

2 Q(x, d) = (l q) T z s T y (1.1) s.t. y = x A T z (1.2) 0 z d (1.3) y 0. (1.4) Observe that the solution of this problem (i.e. the decision variables z and y) depend on the observed value d of the demand as well as on the numbers of parts purchased x. Let Q(x, d) denote the optimal objective value of the LP problem given by (1.1)-(1.4). Now, the optimization problem to find the optimal numbers of parts x (which is to be ordered prior to demand realization) can be formulated as: x 0 ct x + E[Q(x, D)] (1.5) where the expectation is taken with respect to probability distribution of the random demand vector D. The first part of the objective function in (1.5) represents the ordering cost, while the second part represents the expected cost of the optimal production plan, given ordered quantities x. The above is an example of a two-stage stochastic optimization problem. The optimization problems in the first and second stages are given by the formulations (1.5) and (1.1)-(1.4) respectively. As the second-stage problem has to deal with the random demand D, its optimal value Q(x, D) is a random variable. Also, understanding the properties of the second-stage problem is important for solving the first stage problem. Special case: Now, consider the special case of finitely many demand scenarios d 1,, d K occurring with positive probabilities p 1,, p K with K p k = 1. In this case, the above two-stage optimization problem can be formulated as one large-scale LP problem: c T x + p k [(l q) T z k s T y k ] s.t. y k = x A T z k, k = 1,, K 0 z k d k, k = 1,, K y k 0, x 0 k = 1,, K where imization is performed over vector variables x and z k, y k (k = 1,, K). Note that, due to different demand scenarios d k (k = 1,, K), the number of decision variables z k and y k and the number of constraints involving these variables are proportional to the number of scenarios K. 2 Linear Two Stage Problem In this section, we discuss two-stage stochastic linear programg problems of the form c T x + E[Q(x, ξ)] (2.1) x R n s.t. Ax = b (2.2) x 0 (2.3) 2

3 where Q(x, ξ) is the optimal value of the second stage problem which is given by Q(x, ξ) = q T y (2.4) y R m s.t. T x + W y = h (2.5) y 0. (2.6) Here ξ := (q, h, T, W ) are the data of the second stage problem. Some or all of the elements of the vector ξ are random. We denote by Ξ the support of the probability distribution of ξ. The expectation operator in the first stage problem (given by (2.1)-(2.3)) is taken with respect to the probability distribution of ξ. The second stage problem (given by (2.4)-(2.6)) is a linear programg problem and its dual formulation can be written as: max π π T (h T x) (2.7) s.t. W T π q. (2.8) By the theory of duality, the optimal values of the above primal (given by (2.4)-(2.6)) and dual (given by (2.7)-(2.8)) problems are equal, unless both problems are infeasible. Hence, if Π(q) := {π : W T π q} is the non-empty feasible set of the given dual problem, we have Q(x, ξ) = max π Π(q) πt (h T x) Note that the set Π(q) is convex, closed and polyhedral. Now, the following propositions present important properties of Q(, ξ). Proposition 2.1: For any given ξ, the function Q(, ξ) is convex. Moreover, if the set Π(q) is nonempty and the problem given by (2.4)-(2.6) is feasible for at least one x, then the function Q(, ξ) is polyhedral. Proof: See proof of Proposition 2.1 in the book. Proposition 2.2: Suppose that for given x = x 0 and ξ Ξ, the value Q(x 0, ξ) is finite. Then Q(, ξ) is sub-differentiable 1 at x 0 and Q(x 0, ξ) = T T D(x 0, ξ) where D(x 0, ξ) := arg max{π T (h T x 0 ) : W T π q} is the set of optimal solutions of the dual problem (given by (2.7)-(2.8)). (Note that, if the maximum in the dual problem is attained at a unique point for x = x 0, then Q(, ξ) is differentiable at x = x 0.) Proof: See proof of Proposition 2.2 in the book. 2.1 The Expected Recourse Cost for Discrete Distributions Consider the expected recourse cost function φ(x) := E[Q(x, ξ)]. As we have said before, the expectation here is taken with respect to the probability distribution of the random vector ξ. Suppose 1 See sub-derivative on Wikipedia for a good definition of sub-differentiability. 3

4 ξ follows a discrete distribution with a finite support Ξ = {ξ 1,, ξ K }. Also, probability of the realization ξ k = (q k, h k, T k, W k ) is p k (k = 1,, K), where K p k = 1. Hence we have φ(x) = E[Q(x, ξ)] = p k Q(x, ξ k ). For a given x, the expectation E[Q(x, ξ)] is equal to the optimal objective value of the linear programg problem E[Q(x, ξ)] = y 1,,y k p k qk T y k s.t. T k x + W k y k = h k, k = 1,, K y k 0, k = 1,, K. Proposition 2.3: Suppose that the probability distribution of ξ has finite support Ξ = {ξ 1,, ξ K } and that the expected recourse cost φ( ) has a finite value in at least one point x R n. Then the function φ( ) is polyhedral, and for any x 0 dom(φ) φ(x 0 ) = p k Q(x 0, ξ k ) Proof: See proof of Proposition 2.3 in the book. The sub-differential Q(x 0, ξ k ) of the optimal value function of the second-stage problem is described in Proposition 2.2. It then follows from there that the expectation function φ is differentiable at x 0 iff for every ξ = ξ k (k = 1,, K), the dual problem given by (2.7)-(2.8) has a unique optimal solution for x = x 0. Example 2.4 in the book shows how the sub-differential of the recourse cost function is found for an example of the linear two-stage model. Section in the book presents properties of the expected recourse cost for general distributions. 2.2 Optimality Conditions for Discrete Distribution Case Theorem 2.10: Let x be a feasible solution of the two-stage problem (given by (2.1)-(2.3) and (2.4)-(2.6)), i.e., x {x : Ax = b, x 0} and φ( x) is finite. Then x is an optimal solution of this problem iff there exist ˆπ k D( x, ξ k ), k = 1,, K, and ˆµ R m such that p k T kˆπ k + A T ˆµ c, x T (c ) p k Tk T ˆπ k A T ˆµ = 0. Proof: See proof of Theorem 2.10 in the book for a formal proof. However, we can also derive the above optimality conditions by finding the optimality conditions of the equivalent large-scale LP formulation which is given by 4

5 c T x + y 1,,y k p k qk T y k (2.9) s.t. T k x + W k y k = h k, k = 1,, K (2.10) Ax = b (2.11) x 0 (2.12) y k 0, k = 1,, K. (2.13) Let us associate dual variables π k (k = 1,, K) and µ with the constraints (2.10) and (2.11) respectively in the above formulation. Now the dual of the above formulation is max b T µ + µ,π 1,,π k s.t. p k h T k π k (2.14) p k Tk T π k + A T µ c (2.15) W T k π k q k, k = 1,, K. (2.16) For above primal and dual formulations, following are the optimality conditions (also known as complimentary slackness conditions in duality). x ( c ) p k Tk T ˆπ k A T ˆµ = 0 (2.17) ȳ T k (q k W T k ˆπ k ) = 0 (2.18) where ( x, {ȳ k } K ) is the optimal solution to the primal problem (given by (2.9)-(2.13)) and (ˆµ, {ˆπ k} K ) is the solution to the dual problem (given by (2.14)-(2.16)). In this case, it is not difficult to show that ˆπ k D( x, ξ k ) (k = 1,, K). Note that equations (2.16) and (2.18) are to be satisfied for existence of such {ˆπ} K. Hence the relations (2.15) and (2.17) are the optimality conditions of the primal problem given by (2.9)-(2.13). The relation (2.15) is the feasibility condition for multipliers ˆµ and {ˆπ} K. The equation (2.17) must hold for optimality. Note: In this class, we discussed mainly two-stage linear stochastic optimization problems. Also, we considered only the case where the distribution of the random vector ξ is discrete with a finite support. Results involving general distributions for such linear models can be found in the book. Additionally, the book also discusses polyhedral (see Section 2.3) and general (see Section 2.4) types of two-stage models. 5