Gradient Sampling Methods for Nonsmooth Optimization

This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various … Read more

An Improved Flow-based Formulation and Reduction Principles for the Minimum Connectivity Inference Problem

The Minimum Connectivity Inference (MCI) problem represents an NP-hard generalisation of the well-known minimum spanning tree problem and has been studied in different fields of research independently. Let an undirected complete graph and finitely many subsets (clusters) of its vertex set be given. Then, the MCI problem is to find a minimal subset of edges … Read more

A Special Complementarity Function Revisited

Recently, a local framework of Newton-type methods for constrained systems of equations has been developed which, applied to the solution of Karush-KuhnTucker (KKT) systems, enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of … Read more

Golden Ratio Algorithms for Variational Inequalities

The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. … Read more

MSEA.jl: A Multi-Stage Exact Algorithm for Bi-objective Pure Integer Linear Programming in Julia

We present a new exact method for bi-objective pure integer linear programming, the so-called Multi-Stage Exact Algorithm (MSEA). The method combines several existing exact and approximate algorithms in the literature to compute the entire nondominated frontier of any bi-objective pure integer linear program. Each algorithm available in MSEA has multiple versions in the literature. Hence, … Read more

The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities

It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit … Read more

Quantitative Stability of Two-stage Stochastic Linear Programs with Full Random Recourse

In this paper, we use the parametric programming technique and pseudo metrics to study the quantitative stability of the two-stage stochastic linear programming problem with full random recourse. Under the simultaneous perturbation of the cost vector, coefficient matrix and right-hand side vector, we first establish the locally Lipschitz continuity of the optimal value function and … Read more

Shortfall Risk Models When Information of Loss Function Is Incomplete

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years for its potential to quantify more effectively the risk of large losses than conditional value at risk. In this paper we consider the case that the true loss function is unavailable either because it is difficult to be identified or the … Read more

A quadratic penalty algorithm for linear programming and its application to linearizations of quadratic assignment problems

This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is … Read more