Optimality Conditions for Constrained Minimax Optimization

Categories Constrained Nonlinear Optimization Tags , , , , ,

Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered … Read more

Regret in the Newsvendor Model with Demand and Yield Randomness

Categories Robust Optimization Tags , , , , ,

We study the fundamental stochastic newsvendor model that considers both demand and yield randomness. It is usually difficult in practice to describe precisely the joint demand and yield distribution, although partial statistical information and empirical data about this ambiguous distribution are often accessible. We combat the issue of distributional ambiguity by taking a data-driven distributionally … Read more

An algorithm for assortment optimization under parametric discrete choice models

Categories 0-1 Programming, Applications - OR and Management Sciences Tags , , ,

This work concerns the assortment optimization problem that refers to selecting a subset of items that maximizes the expected revenue in the presence of the substitution behavior of consumers specified by a parametric choice model. The key challenge lies in the computational difficulty of finding the best subset solution, which often requires exhaustive search. The … Read more

Mixed-Integer Nonlinear Optimization for District Heating Network Expansion

Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering Tags , ,

We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the … Read more

A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

Categories (Mixed) Integer Linear Programming, Game Theory, Stochastic Programming Tags , , , , ,

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

DMulti-MADS: Mesh adaptive direct multisearch for blackbox multiobjective optimization

Categories Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to constrained multiobjective derivative-free optimization. This method … Read more

A Framework for Generalized Benders’ Decomposition and Its Application to Multilevel Optimization

Categories (Mixed) Integer Linear Programming, Game Theory, Stochastic Programming Tags , , ,

We describe an algorithmic framework generalizing the well-known framework originally introduced by Benders. We apply this framework to several classes of optimization problems that fall under the broad umbrella of multilevel/multistage mixed integer linear optimization problems. The development of the abstract framework and its application to this broad class of problems provides new insights and … Read more

High Dimensional Three-Periods Locally Ideal MIP Formulations for the UC Problem

Categories (Mixed) Integer Nonlinear Programming, Energy, Scheduling Tags , , , ,

The thermal unit commitment (UC) problem often can be formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. The tighter characteristic re-duces the search space, therefore, as a natural conse-quence, significantly reduces the computational burden. In the literature, many tightened formulations for single units with parts … Read more

On Linear Optimization over Wasserstein Balls

Categories Robust Optimization, Stochastic Programming Tags , , , ,

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under … Read more

Convex Hull Representations for Bounded Products of Variables

Categories Convex Optimization, Global Optimization Theory, Second-Order Cone Programming Tags , , ,

It is well known that the convex hull of {(x,y,xy)}, where (x,y) is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti et al. (2010) and Miller et al. (2011) showed that if there are additional upper and/or lower bounds on the product z=xy, then the convex hull can … Read more