A Computational Study for Piecewise Linear Relaxations of Mixed-Integer Nonlinear Programs

Solving mixed-integer nonlinear problems by means of piecewise linear relaxations can be a reasonable alternative to the commonly used spatial branch-and-bound. These relaxations have been modeled by various mixed-integer models in recent decades. The idea is to exploit the availability of mature solvers for mixed-integer problems. In this work, we compare different reformulations in terms … Read more

Optimization and Simulation for the Daily Operation of Renewable Energy Communities

Renewable Energy Communities (RECs) are an important building block for the decarbonization of the energy sector. The concept of RECs allows individual consumers to join together in local communities to generate, store, consume and sell renewable energy. A major benefit of this collective approach is a better match between supply and demand profiles, and thus, … Read more

K-Shortest Simple Paths Using Biobjective Path Search

In this paper we introduce a new algorithm for the k-Shortest Simple Paths (k-SSP) problem with an asymptotic running time matching the state of the art from the literature. It is based on a black-box algorithm due to Roddity and Zwick that solves at most 2k instances of the Second Shortest Simple Path (2-SSP) problem … Read more

Continuous Selections of Solutions to Parametric Variational Inequalities

This paper studies the existence of a (Lipschitz) continuous (single-valued) solution function of parametric variational inequalities under functional and constraint perturbations. At the most elementary level, this issue can be explained from classical parametric linear programming and its resolution by the parametric simplex method, which computes a solution trajectory of the problem when the objective … Read more

Solving Hard Bi-objective Knapsack Problems Using Deep Reinforcement Learning

We study a class of bi-objective integer programs known as bi-objective knapsack problems (BOKPs). Our research focuses on the development of innovative exact and approximate solution methods for BOKPs by synergizing algorithmic concepts from two distinct domains: multi-objective integer programming and (deep) reinforcement learning. While novel reinforcement learning techniques have been applied successfully to single-objective … Read more

Functions associated with the nonconvex second-order cone

The nonconvex second-order cone (nonconvex SOC for short) is a nonconvex extension to the convex second-order cone, in the sense that it consists of any vector divided into two sub-vectors for which the Euclidean norm of the first sub-vector is at least as large as the Euclidean norm of the second sub-vector. This cone can … Read more

Preconditioning for Generelized Jacobians with the ω-Condition Number

Preconditioning is essential in iterative methods for solving linear systems of equations. We study a nonclassic matrix condition number, the ω-condition number, in the context of optimal conditioning for low rank updating of positive definite matrices. For a positive definite matrix, this condition measure is the ratio of the arithmetic and geometric means of the … Read more

Eigenvalue programming beyond matrices

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the “eigenvalues” of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret “eigenvalues” in a broad fashion going beyond the usual eigenvalues of matrices. This … Read more

Cone product reformulation for global optimization

In this paper, we study nonconvex optimization problems involving sum of linear times convex (SLC) functions as well as conic constraints belonging to one of the five basic cones, that is, linear cone, second order cone, power cone, exponential cone, and semidefinite cone. By using the Reformulation Perspectification Technique, we can obtain a convex relaxation … Read more

Range of the displacement operator of PDHG with applications to quadratic and conic programming

Primal-dual hybrid gradient (PDHG) is a first-order method for saddle-point problems and convex programming introduced by Chambolle and Pock. Recently, Applegate et al. analyzed the behavior of PDHG when applied to an infeasible or unbounded instance of linear programming, and in particular, showed that PDHG is able to diagnose these conditions. Their analysis hinges on … Read more