Dual solutions in convex stochastic optimization

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental … Read more

Distributionally Robust Modeling of Optimal Control

The aim of this paper is to formulate several questions related to distributionally robust Stochastic Optimal Control modeling. As an example, the distributionally robust counterpart of the classical inventory model is discussed in details. Finite and infinite horizon stationary settings are considered. Article Download View Distributionally Robust Modeling of Optimal Control

Strong duality of a conic optimization problem with a single hyperplane and two cone constraints

Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view … Read more

Linearizing Bilinear Products of Shadow Prices and Dispatch Variables in Bilevel Problems for Optimal Power System Planning

This work presents a general method for linearizing bilinear terms in the upper level of bilevel optimization problems when the bilinear terms are products of the primal and dual variables of the lower level. Bilinear terms of this form often appear in energy market optimization models where the dual variable represents the market price of … Read more

Dual SDDP for risk-averse multistage stochastic programs

Risk-averse multistage stochastic programs appear in multiple areas and are challenging to solve. Stochastic Dual Dynamic Programming (SDDP) is a well known tool to address such problems under time-independence assumptions. We show how to derive a dual formulation for these problems and apply an SDDP algorithm, leading to converging and deterministic upper bounds for risk-averse … Read more

Distributionally Robust Optimal Control and MDP Modeling

In this paper, we discuss Optimal Control and Markov Decision Process (MDP) formulations of multistage optimization problems when the involved probability distributions are not known exactly, but rather are assumed to belong to specified ambiguity families. The aim of this paper is to clarify a connection between such distributionally robust approaches to multistage stochastic optimization. … Read more

Bound Propagation for Linear Inequalities Revisited

In 2011, Korovin and Voronkov (Proceedings of the 23rd International Conference on Automated Deduction, vol. 6803 of Lecture Notes in Computer Science, pp. 369-383) proposed a method based on bound propagation for solving systems of linear inequalities. In this paper, an alternate description of their algorithm which also incorporates an addition that returns a certificate … Read more

On the strong concavity of the dual function of an optimization problem

We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the the assumption of strong convexity of the objective cannot be weakened to convexity and that the assumption that the gradients of all constraints at the optimal solution are … Read more

A simplified treatment of Ramana’s exact dual for semidefinite programming

In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. … Read more

A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, … Read more