Global Optimization

Solving Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification – a Case Study for Gas Networks

Categories Applications - Science and Engineering, Global Optimization, Network Optimization Tags , , ,

Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each … Read more

Pump scheduling in drinking water distribution networks with an LP/NLP-based branch and bound

Categories Global Optimization Applications Tags , , ,

This paper offers a novel approach for computing globally optimal solutions to the pump scheduling problem in drinking water distribution networks. A tight integer linear relaxation of the original non-convex formulation is devised and solved by branch and bound where integer nodes are investigated through non-linear programming to check the satisfaction of the non-convex constraints … Read more

Sparse PSD approximation of the PSD cone

Categories Global Optimization Theory, Semi-definite Programming Tags ,

While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k times k principal submatrices — we call this the sparse SDP relaxation. Surprisingly, … Read more

On convex hulls of epigraphs of QCQPs

Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming Tags , , , ,

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such … Read more

On the algebraic structure of the copositive cone

Categories Applications - OR and Management Sciences, Global Optimization Theory Tags , , ,

We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is … Read more

A new discrete filled function with generic local searches for global nonlinear integer optimization

Categories Global Optimization Applications, Global Optimization Theory Tags , , ,

The problem of finding global minima of nonlinear discrete functions arises in many fields of practical matters. In recent years, methods based on discrete filled functions become popular as ways of solving these sort of problems. However, they rely on the steepest descent method for local searches. Here we present an approach that does not … Read more

Nonconvex Constrained Optimization by a Filtering Branch and Bound

Categories Constrained Nonlinear Optimization, Global Optimization Theory, Multi-Criteria Optimization

A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the alphaBB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained … Read more

A dimensionality reduction technique for unconstrained global optimization of functions with low effective dimensionality

Categories Global Optimization, Stochastic Approaches Tags , , ,

We investigate the unconstrained global optimization of functions with low effective dimensionality, that are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in [Wang et al., Bayesian optimization in a billion dimensions via random embeddings. JAIR, 55(1): 361–387, 2016], we study a generic Random Embeddings for Global Optimization (REGO) framework … Read more

Stochastic Dual Dynamic Programming for Multistage Stochastic Mixed-Integer Nonlinear Optimization

Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Stochastic Programming Tags , , ,

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with \emph{non-Lipschitz-continuous} value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient … Read more