When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems. Electronic supplementary material The online version of this article (doi:10.1007/s10107-017-1166-z) contains supplementary material, which is available to authorized users.

The contribution of this paper is twofold: 1. We prove for convex problems with concave uncertainty, which also satisfy a set of other conditions, that the objective values of the corresponding RO and ARO problems are equal. This is an extension of the result in [6, Proposition 2.1], which is only for problems that are linear in the variables and uncertain parameters. 2. We study uncertain nonlinear problems in which some of the uncertain parameters are constraint-wise and the rest are not. In particular, we prove that for an ARO problem, under a set of conditions similar to the pure constraint-wise cases, there is an optimal decision rule that depends only on the non-constraint-wise uncertain parameters. Moreover, we show that for a specific class of problems, there is an affine decision rule that is only a function of the non-constraint-wise uncertain parameters and that yields the same objective value as using an affine decision rule that is a function of all uncertain parameters. The first contribution means that for this class of problems, there is no need to solve ARO ones. This has two outstanding merits: first, solving an RO problem is computationally much easier than solving an ARO one; and second, since ARO is an online approach, parts of the solution for a problem can only be implemented once the values of the uncertain parameters are known. The RO approach is an offline one, however, so all preparations for implementing the solution can start immediately upon solving the RO problem (for further discussion about online and offline approaches see [19]).
The merit of the second contribution is that it reduces the number of variables in the problem by using affine decision rules, since we know beforehand that there is an optimal solution for ARO where the coefficients of the constraint-wise uncertain parameters are zero.
In the last part of the paper, we apply our theoretical results to an important class of problems. We show that our contributions are applicable to convex quadratic and/or conic quadratic problems, which can arise in multi-stage portfolio optimization, for example.
We emphasize that the results obtained in this paper concern the worst-case objective value of an ARO problem. We provide conditions under which the optimal RO solutions are also optimal for the ARO problem. However, in such cases, another ARO optimal solution may yield a better average-case objective value [17].
The rest of the paper is organized as follows: Sect. 2 presents our main results. We provide a set of conditions under which constraint-wise RO and ARO problems have the same optimal objective values. Moreover, for problems in which just some of the uncertain parameters are constraint-wise and not all, we show that under a similar set of conditions, there is an optimal decision rule that is independent of the constraintwise uncertain parameters. In Sect. 3, we apply our results to convex quadratic and conic quadratic problems. set of conditions for problems with constraint-wise uncertainty under which adjustable and static robust optimization produces the same optimal values. In Sect. 2.3 we study problems in which only some of the uncertain parameters are constraint-wise and the rest are not.

Preliminaries
Consider the following uncertain nonlinear optimization problem where ζ ∈ Z ⊆ R l is an uncertain parameter and Z is a nonempty uncertainty set, x ∈ X ⊆ R r is a non-adjustable variable, and X is a nonempty set defined by constraints that depend only on x, y ∈ Y(x) ⊆ R n is an adjustable variable and Y(x) is defined by constraints independent of ζ . Also, we assume that f (ζ, x, y) and g i (ζ, x, y), i = 1, . . . , m, are continuous.
We can define static and adjustable robust optimization problems corresponding to uncertain problem (1).

Definition 1 (Static robust optimization)
For problem (1), the static robust counterpart (RC) is defined by Definition 2 (Adjustable robust optimization) For problem (1), there are two different definitions for the adjustable robust counterpart (ARC): and The equivalence of problems (2) and (ARC) is proved in [20]. We denote the objective values of problems (RC) and (ARC) by Opt (RC) and Opt (ARC), respectively. We extend the definition of (ARC) with fixed recourse for a linear problem with linear uncertainty in [4] to the nonlinear case (nonlinear problem with nonlinear uncertainty) in the following definition. Definition 3 (Fixed recourse problem) (ARC) has fixed recourse when there are continuous functionsf ,g i : R n+r → R,f ,ḡ i : R l+r → R, for i = 1, . . . , m, such that for all ζ ∈ Z ⊂ R l , x ∈ X ⊂ R r , and y ∈ Y(x) ⊂ R n , In this paper, we work primarily with constraint-wise uncertainty, which is defined as follows.
Definition 4 (Constraint-wise uncertainty [4]) For problem (1), the uncertainty is constraint-wise when each uncertain parameter ζ can be split into blocks ζ = [ζ 0 , . . . , ζ m ] ∈ R l such that the data of the objective depends only on ζ 0 ∈ R l 0 , the data of the i-th constraint depends solely on ζ i ∈ R l i , and the uncertainty set Constraint-wise uncertainty appears in Markov decision process, for example, where it is called rectangularity [18,21].
Notice that problem (1) does not contain any equality constraint that depends on ζ . The usual way of dealing with such uncertain equalities in (ARC) is to eliminate adjustable variables [14,Section 7]. This means that we are implicitly forcing the adjustable variables that are eliminated to obey specific decision rules. This is not allowed in (RC). We illustrate this in Example 4 of Supplementary Material Section D.
We will now outline the assumptions used in this paper to express conditions under which Opt (RC) = Opt (ARC).

Assumption 1
All the assumptions are with respect to problem (1). Throughout this paper, we assume that i. There is no equality constraint in problem (1) that depends on ζ . ii. The uncertainty set Z is compact.
Assumptions i, iii, iv, vi, and vii are essentially the framework of static robust convex optimization considered in [5].

Constraint-wise uncertainty
In this subsection, we study problems with constraint-wise uncertainty and provide a set of conditions, under which Opt (RC) and Opt (ARC) are equal.

Theorem 1 If problem (1) has constraint-wise uncertainty and Assumptions i-vii hold, then O pt (RC) = Opt (ARC).
Proof The line of reasoning is the same as in [4, Theorem 2.1].
Case I: Suppose that (ARC) does not have a non-adjustable variable. First, we assume that (RC) is feasible. So, it is sufficient to show that when- According to the definitions, we have: and, If Y = ∅, it is clear that Opt (ARC) = Opt (RC) = +∞. Now, assume that Y = ∅. By contradiction, suppose that there is a scalart such thatt ≥ Opt (ARC) andt < Opt (RC). Because of the constraint-wise uncertainty, . . , m, and by (4), it follows that Also, continuity implies where U y is the intersection of a 2-norm open ball with a strictly positive where = min k y k . As a simplification, we set ζ k = ζ y k i y k , i k = i y k and Since Y is convex and all f k (z) are convex and continuous on Y due to Assumption vii, and because max k f k (z) ≥ for each z ∈ Y, there are nonnegative weights λ k with k λ k = 1 such that We define It is clear by convexity of Z thatζ ∈ Z. Additionally, due tot ≥ Opt (ARC), we have which means Also, we know that for each i = 0, . . . , m, the functions G i (ζ i ,ȳ) are concave in ζ i due to Assumption vi. Hence, for all i = 0, . . . , m, and w i > 0

Summing over the indices results in
By applying (7) and (10), the above inequality contradicts > 0. Now we consider the case where (RC) is not feasible, which means Opt (RC) = +∞. To prove equality of (RC) and (ARC) with respect to the worst-case objective value, it is sufficient to show that there is not ∈ R such thatt ≥ Opt (ARC). So, the same argument used in the previous part implies that Opt (ARC) = +∞. Case II: Now, we consider a general case, where (ARC) contains the non-adjustable variable x. As proved in Case I, for any x ∈ X , and have the same optimal value. Therefore, taking the infimum over all x ∈ X results in Opt (RC) = Opt (ARC).

Non-constraint-wise uncertainty
Section 2.2 focuses on constraint-wise uncertainty. The question is what can be concluded for a problem in which some, but not all, of the uncertain parameters are constraint-wise. Consider the following problem: where ζ = (ζ 0 , . . . , ζ m ) ∈ Z = Z 0 × . . . × Z m and α ∈ A ⊆ R d are uncertain parameters (ζ is constraint-wise and α is non-constraint-wise). This problem has a hybrid uncertainty, so we cannot use the results in Section 2.2 to deduce equality of the optimal values of the hybrid robust counterpart (H RC) and the corresponding hybrid adjustable robust counterpart (H ARC). However, the following corollary states that if in such a case the same set of conditions as in Theorem 1 hold with respect to the constraint-wise uncertain parameters, then there exists an optimal decision rule that is a function of only the non-constraint-wise uncertain parameters. In other words, the two problems s. t. f (ζ 0 , α, x, y(ζ, α)) ≤ t (ζ, α), have the same optimal objective values. We denote the optimal objective values of (H ARC) and (H ARC α ) by Opt (H ARC) and Opt (H ARC α ), respectively.

Corollary 1 Suppose that for all α ∈ A, the assumptions of Theorem 1 hold with respect to ζ, x, y. Then, O pt (H ARC) = Opt (H ARC α ).
Proof By fixing α ∈ A and x ∈ X and applying Theorem 1, the optimal objective value of are equal. The result follows from taking the supremum over α ∈ A and infimum over x ∈ X .
Corollary 1 can be used to reduce the complexity of solving adjustable robust optimization problems. This is because in order to solve (H ARC), one needs to find an optimal decision rule with respect to both α and ζ , but, we can ensure the existence of an optimal decision rule that only depends on α by applying this corollary.
It is important to note that if we restrict ourselves to one class of decision rules, e.g., affine decision rules, as is customary, then Corollary 1 does not necessarily guarantee the existence of an optimal affine decision rule that only depends on α. The following corollary, however, states that if the problem has fixed recourse with respect to the constraint-wise uncertain parameter ζ and we use a specific class of decision rules that are separable with respect to ζ and α, then there exists an optimal decision rule that depends only on α.
Let us denote byȳ ω (α): R d → R a function of α that belongs to a specific class parametrized by ω. One of the examples forȳ ω (α) is a polynomial. In this case, ω could be the vector of coefficients for the monomials.
In Corollary 2, y(ζ ) is a general function. For instance, if we assume thatȳ ω (α) lies in the class of affine functions, even for a general y(ζ ), the optimal objective value is independent from ζ . The other example is when both y(ζ ) andȳ ω (α) are affine, which means that the decision rule is affine. We consider this case in the next corollary. (14). Then, using an affine decision rule, y(α) = u + W α or y(ζ, α) = u + V ζ + W α, where u ∈ R n , V ∈ R n×l , and W ∈ R n×d , yields the same approximate optimal value. Corollary 3 mentions two different problems for approximating (H ARC): one considers y(α) = u + W α as the form of decision rule and the other y(ζ, α) = u + V ζ + W α. We denote the optimal objective values of the former affinely adjustable robust counterparts by Opt (A ARC α ) and Opt (A ARC ζ,α ), respectively. Then, in general, for problem (H RC), we have

Corollary 3 Suppose that in (H A RC) the constraints and objective functions satisfy
In this section, we discussed conditions that turn inequalities in (19) into equalities. Theorem 1 provides a set of conditions under which all of the inequalities can be replaced by equalities. In addition, under a similar set of conditions as in that theorem, Corollary 3 ensures us that the middle inequality in (19) turns into an equality. Other sets of conditions for which Opt (H ARC) = Opt (A ARC ζ,α ) are proposed in [8,11,16]. In Supplementary Material Section D, we provide some examples to show that these inequalities can be strict.

Application
One application of the results derived in Sect. 2 is for the following problem: where g j (α, x, y), j = 1, . . . , m, is a continuous function, the convex quadratic function h i is defined as and the conic quadratic function p k is defined as where α ∈ R l , ζ i ∈ R l i , and θ k ∈ R l I +k are the uncertain parameters for some integers l, l i , l I +k , i = 1, . . . , I , k = 1, . . . , K , and x and y are non-adjustable and adjustable variables, respectively. We assume that the matrices A i (ζ i ) and B k (θ k ) are positive semi-definite for all ζ i ∈ Z i and θ k ∈ T k , i = 1, . . . , I , k = 1, . . . , K . Also, we assume that , and e k (θ k ) are affine in ζ i and θ k , i = 1, . . . , I , k = 1, . . . , K , respectively. This type of problem arises, for example, when a part of the problem is related to multi-stage mean-variance portfolio optimization [15], in which the asset return mean and covariance matrix are uncertain and these uncertainties only occur in the objective function (hence the problem has constraint-wise uncertainty).
If the uncertainty over α is constraint-wise and g j (α, x, y) is concave in α and convex in y, j = 1, . . . , m; A, Z i and T k are convex, i = 1, . . . , I , k = 1, . . . , K ; and Y(x) is compact and convex for all x ∈ X , then by Theorem 1, the optimal values of the corresponding static and adjustable robust problems are equal, because h i and p k are convex in y and concave in ζ i and θ k , i = 1, . . . , I , k = 1, . . . , K , respectively. Moreover, if the uncertainty over α is not constraint-wise, then by Corollary 1, an optimal y exists for the corresponding adjustable robust counterpart that is independent of ζ i and θ k , i = 1, . . . , I , k = 1, . . . , K .
variables and concave with respect to the uncertain parameters, and that have a convex compact uncertainty set and adjustable variables that lie in a convex compact set.
This result does not hold for problems where just some of the uncertain parameters are constraint-wise. We prove that under a set of assumptions similar to the pure constraint-wise case, there exists an optimal decision rule that does not depend on the constraint-wise uncertain parameters. Also, we show that for one class of problems, restricting decision rules to be affine and independent of the constraint-wise uncertain parameters yields the same optimal objective value as in cases where the decision rules are affine and dependent on both the constraint-wise and non-constraint-wise uncertain parameters.
Lastly, we prove that for adjustable robust optimization problems with convex quadratic and/or conic quadratic constraints, if the uncertainty in the quadratic constraints is constraint-wise, then optimal adjustable variables exist that are independent of the constraint-wise uncertain parameters.