Duality for Robust Linear Infinite Programming Problems Revisited

Abstract

In this paper, we consider the robust linear infinite programming problem (RLIP_c) defined by

$$ \begin{array}{@{}rcl@{}} (\text{RLIP}_{c})\quad &&\inf \langle c,x\rangle \\ \text{subject to } &&x\in X,~\langle x^{\ast},x \rangle \le r,~\forall (x^{\ast},r)\in\mathcal{U}_{t},~ \forall t\in T, \end{array} $$

where X is a locally convex Hausdorff topological vector space, T is an arbitrary index set, c ∈ X^∗, and $\mathcal {U}_{t}\subset X^{\ast }\times \mathbb {R}$, t ∈ T are uncertainty sets. We propose an approach to duality for the robust linear problems with convex constraints (RP_c) and establish corresponding robust strong duality and also, stable robust strong duality, i.e., robust strong duality holds “uniformly” with all c ∈ X^∗. With the different choices/ways of setting/arranging data from (RLIP_c), one gets back to the model (RP_c) and the (stable) robust strong duality for (RP_c) applies. By such a way, nine versions of dual problems for (RLIP_c) are proposed. Necessary and sufficient conditions for stable robust strong duality of these pairs of primal-dual problems are given, for which some cover several known results in the literature while the others, due to the best knowledge of the authors, are new. Moreover, as by-products, we obtained from the robust strong duality for variants pairs of primal-dual problems, several robust Farkas-type results for linear infinite systems with uncertainty. Lastly, as extensions/applications, we extend/apply the results obtained to robust linear problems with sub-affine constraints, and to linear infinite problems (i.e., (RLIP_c) with the absence of uncertainty). It is worth noticing even for these cases, we are able to derive new results on (robust/stable robust) duality for the mentioned classes of problems and new robust Farkas-type results for sub-linear systems, and also for linear infinite systems in the absence of uncertainty.

Appendices

Appendix A: Proof of Proposition 6

• (i) From the proof of Theorem 2 for the case i = 1, we can see that the problem (RLIP_c) can be transformed to (RP_c) with $Z=\mathbb {R}$, $S=\mathbb {R}_{+}$, $\mathcal {U}={\mathcal{V}}$, G_v(⋅) = v¹(⋅) − v² for all $v\in {\mathcal{V}}$, and in such a case, ${\mathcal{M}}_{0}=\mathcal {N}_{1}$. Observe that the functions v↦〈v¹, x〉− v₂, x ∈ X, are concave (actually, they are affine). Together with the fact that ${\mathcal{V}}$ is convex and $Z=\mathbb {R}$, the collection (v↦G_v(x))_x∈X is uniformly $\mathbb {R}_{+}$-concave. So, in the light of Proposition 2(i), ${\mathcal{M}}_{0}$ is convex, and so is $\mathcal {N}_{1}$.

• (ii) By the same argument as above, to prove $\mathcal {N}_{3}$ is convex, it is sufficient to show that the collection (u↦G_u(x))_x∈X is uniformly $\mathbb {R}_{+}^{(T)}$-concave with $\mathcal {U}={\mathcal{U}}$, $Z=\mathbb {R}^{T}$ and $G_{u}(\cdot )=(\langle {u^{1}_{t}},\cdot \rangle - {u^{2}_{t}})_{t\in T}$ for all $u\in {\mathcal{U}}$ (the setting in the proof of Theorem 2 for the case i = 3). Now, take arbitrarily $\lambda ,\mu \in \mathbb {R}^{(T)}_{+}$ and $u,w\in {\mathcal{U}}$. Let $\bar \lambda \in \mathbb {R}^{(T)}_{+}$ and $\bar u\in {\mathcal{U}}$ such that $\bar \lambda _{t}=\lambda _{t}+\mu _{t}$, ${\bar {u}^{2}_{t}}=\min \limits \{{u^{2}_{t}}, {w^{2}_{t}}\}$ and

  $$ {\bar{u}_{t}^{1}}=\left\{\begin{array}{ll} \frac{1}{\bar \lambda_{t}}(\lambda_{t} {u^{1}_{t}}+\mu_{t} {w^{1}_{t}})&\quad \text{ if } \lambda_{t}+\mu_{t}\ne 0,\\ {u^{1}_{t}}&\quad\text{ otherwise} \end{array}\right. $$

  ($\bar u \in {\mathcal{U}}$ as $\{x^{\ast }\in X^{\ast }: (x^{\ast }, r)\in \mathcal {U}_{t}\}$ is convex for all t ∈ T). Then, it is easy to check that

  $$ \lambda_{t}(\langle {u^{1}_{t}},x\rangle -{u^{2}_{t}}) + \mu_{t}(\langle {w^{1}_{t}},x\rangle - {w^{2}_{t}}) \le \bar \lambda_{t} (\langle {\bar{u}^{1}_{t}},x\rangle - {\bar{u}^{2}_{t}})\quad\forall t\in T,~\forall x\in X, $$

  and consequently,

  $$ \sum\limits_{t\in T}{\lambda^{1}_{t}}(\langle {u^{1}_{t}},x\rangle - {u^{2}_{t}}) + \sum\limits_{t\in T}{\lambda^{2}_{t}}(\langle {w^{1}_{t}},x\rangle - {w^{2}_{t}})\le \sum\limits_{t\in T}\bar \lambda_{t} (\langle {\bar{u}^{1}_{t}},x\rangle - {\bar{u}^{1}_{t}})\quad \forall x\in X, $$

  which means $\lambda G_{u}(x)+\mu G_{w}(x)\le \bar \lambda G_{\bar u}(x)$ for all x ∈ X, yielding the uniform $\mathbb {R}_{+}^{(T)}$-concavity of the collection (u↦G_u(x))_x∈X. The conclusion now follows from Proposition 2(i).

• (iii) Recall that $\mathcal {N}_{4}$ is a specific form of ${\mathcal{M}}_{0}$ with $Z=\mathbb {R}$, $S=\mathbb {R}_{+}$, $\mathcal {U}=T$, and $G_{t}(\cdot )= \sup _{v\in \mathcal {U}_{t}} [\langle v^{1}, \cdot \rangle - v^{2}]$ for all t ∈ T (the setting in the proof of Theorem 2 for the case i = 4). Now, for each t ∈ T and x ∈ X, as $\mathcal {U}_{t}=\mathcal {U}^{1}_{t}\times \mathcal {U}^{2}_{t}$ (with $\mathcal {U}^{1}_{t}\subset X^{\ast }$ and $\mathcal {U}^{2}_{t} \subset \mathbb {R}$), it holds

  $$ G_{t}(x)=\sup\limits_{x^{\ast}\in \mathcal{U}^{1}_{t}} \langle x^{\ast},x\rangle - \inf\limits_{r\in \mathcal{U}^{2}_{t}}r = \sup\limits_{x^{\ast}\in \mathcal{U}^{1}_{t}} \langle x^{\ast},x\rangle - \inf \mathcal{U}^{2}_{t}. $$

  So, for all x ∈ X, because T is convex, $t\mapsto \sup _{x^{\ast }\in \mathcal {U}^{1}_{t}} \langle x^{\ast }, x\rangle$ is affine, and $t \mapsto \inf \mathcal {U}^{2}_{t}$ is convex, the function t↦G_t(x) is concave. This accounts for the uniform $\mathbb {R}^{(T)}_{+}$-concavity of the collection (t↦G_t(x))_x∈X. The conclusion again follows from Proposition 2(i).

• (iv) Consider the ways of transforming (RLIP_c) to (RP_c) in the proofs of Theorem 3 for the case i = 6, 7. Note that, in these ways, the uncertain set $\mathcal {U}$ is always a singleton. So, the corresponding qualifying sets (i.e, $\mathcal {N}_{6}$ and $\mathcal {N}_{7}$) are always convex (see Remark 3).

Appendix B: Proof of Proposition 7

Recall that $\mathcal {N}_{i}$, i = 1, 2,…, 7, are specific forms of ${\mathcal{M}}_{0}$ following the corresponding ways transforming of (RLIP_c) to (RP_c) considered in the proofs of Theorems 2 and 3. So, to prove that $\mathcal {N}_{i}$ is closed, we make use of Proposition 2(ii), which provides some sufficient condition for the closedness of the robust moment cone ${\mathcal{M}}_{0}$.

• (i) For i = 1, let us consider the way of transforming (RLIP_c) to (RP_c) by setting $Z=\mathbb {R}$, $S=\mathbb {R}_{+}$, $\mathcal {U}={\mathcal{V}}$, and G_v(⋅) = 〈v¹,⋅〉− v² for all $v \in {\mathcal{V}}$. For all x ∈ X, it is easy to see that the function v↦G_v(x) = 〈v¹, x〉− v² is continuous, and hence, it is $\mathbb {R}^{+}$-usc (see Remark 1(iii)). Moreover, $\text {gph} {\mathcal{U}}$ is compact, $\mathbb {R}$ is normed space, and (16) ensures the fulfilling of condition (C₀) in Proposition 2. The closedness of $\mathcal {N}_{1}$ follows from Proposition 2(ii).

• (ii) For i = 4, consider the way of transforming with the setting $Z=\mathbb {R}$, $S=\mathbb {R}_{+}$, $\mathcal {U}=T$, and $G_{t}(\cdot )= \sup _{v\in \mathcal {U}_{t}} [\langle v^{1}, \cdot \rangle - v^{2}]$ for all t ∈ T. One has that $\mathcal {U}=T$ is a compact set, that $t\mapsto G_{t}(x)= \sup _{v\in \mathcal {U}_{t}} [\langle v^{1}, x\rangle -v^{2}]$ is usc and hence, it is $\mathbb {R}^{+}$-usc, and that Slater-type condition (C₀) holds (as (17) holds). The conclusion now follows from Proposition 2(ii).

• (iii) Consider the way of transforming which corresponds to i = 5, i.e., we consider $Z=\mathbb {R}$, $S=\mathbb {R}_{+}$, $\mathcal {U}={\mathcal{U}}$, and $G_{u}(\cdot )= \sup _{t\in T}[\langle {u^{1}_{t}}, \cdot \rangle - {u^{2}_{t}}]$ for all $u\in {\mathcal{U}}$. As ${\mathcal{U}}={\prod }_{t\in T} \mathcal {U}_{t}$, the assumption that $\mathcal {U}_{t}$ is compact for all t ∈ T which entails the compactness of ${\mathcal{U}}$. The other assumptions ensure the fulfillment of conditions in Proposition 2(ii) and the conclusion follows from this very proposition.

• (iv) For i = 7, using the same argument as above in transforming (RLIP_c) to (RP_c) in the proof of Theorem 3. As by this way, the uncertainty set is a singleton, and hence, $\mathcal {N}_{7}$ is convex (see Remark 3). Now from Proposition 2(ii), Slater-type condition ensures the closedness of the robust moment cone $\mathcal {N}_{7}$, as desired.