Globally Convergent Derivative-Free Methods in Nonconvex Optimization with and without Noise

This paper addresses the study of nonconvex derivative-free optimization problems, where only information of either smooth objective functions or their noisy approximations is available. General derivative-free methods are proposed for minimizing differentiable (not necessarily convex) functions with globally Lipschitz continuous gradients, where the accuracy of approximate gradients is interacting with stepsizes and exact gradient values. … Read more

BattOpt: Optimal Facility Planning for Electric Vehicle Battery Recycling

The electric vehicle (EV) battery supply chain will face challenges in sourcing scarce, expensive minerals required for manufacturing and in disposing of hazardous retired batteries. Integrating recycling technology into the supply chain has the potential to alleviate these issues; however, players in the battery market must design investment plans for recycling facilities. In this paper, … Read more

MUSE-BB: A Decomposition Algorithm for Nonconvex Two-Stage Problems using Strong Multisection Branching

\(\) We present MUSE-BB, a branch-and-bound (B&B) based decomposition algorithm for the deterministic global solution of nonconvex two-stage stochastic programming problems. In contrast to three recent decomposition algorithms, which solve this type of problem in a projected form by nesting an inner B&B in an outer B&B on the first-stage variables, we branch on all … Read more

Nonconvex optimization problems involving the Euclidean norm: Challenges, progress, and opportunities

The field of global optimization has advanced significantly over the past three decades. Yet, the solution of even small instances of many nonconvex optimization problems involving the Euclidean norm to global optimality remains beyond the reach of modern global optimization methods. These problems include numerous well-known and high-impact open research questions from a diverse collection … Read more

Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: 1) … Read more

Spatial branch-and-bound for nonconvex separable piecewise linear optimization

Nonconvex separable piecewise linear functions (PLFs) frequently appear in applications and to approximate nonlinearitites. The standard practice to formulate nonconvex PLFs is from the perspective of discrete optimisation, using special ordered sets and mixed integer linear programs (MILPs). In contrast, we take the viewpoint of global continuous optimization and present a spatial branch-and-bound algorithm (sBB) … Read more

Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization

We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more

Revisiting the fitting of the Nelson-Siegel and Svensson models

The Nelson-Siegel and the Svensson models are two of the most widely used models for the term structure of interest rates. Even though the models are quite simple and intuitive, fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, a novel mathematical analysis of the fitting problem … Read more

The best approximation pair problem relative to two subsets in a normed space

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. This problem, which has a long history, has … Read more