Semidefinite Programming and Nash Equilibria in Bimatrix Games

We explore the power of semidefinite programming (SDP) for finding additive epsilon-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is … Read more

A Benders squared (B2) framework for infinite-horizon stochastic linear programs

We propose a nested decomposition scheme for infinite-horizon stochastic linear programs. Our approach can be seen as a provably convergent extension of stochastic dual dynamic programming to the infinite-horizon setting: we explore a sequence of finite-horizon problems of increasing length until we can prove convergence with a given confidence level. The methodology alternates between a … Read more

A Mixed Integer Programming Model to Analyse and Optimise Patient Flow in a Surgical Suite.

Demand for healthcare services is growing rapidly in Australia and across the world, and rising healthcare expenditure is increasing pressure on sustainability of government-funded healthcare systems. In Australia, elective surgery waiting lists are growing and hospitals are struggling with a capacity shortage. To keep up with the rising demand, we need to be more efficient … Read more

On the Linear Extension Complexity of Stable Set Polytopes for Perfect Graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we … Read more

SDP-based Branch-and-Bound for Non-convex Quadratic Integer Optimization

Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed [11], which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST … Read more

Properties of the block BFGS update and its application to the limited-memory block BNS method for unconstrained minimization.

A block version of the BFGS variable metric update formula and its modifications are investigated. In spite of the fact that this formula satisfies the quasi-Newton conditions with all used difference vectors and that the improvement of convergence is the best one in some sense for quadratic objective functions, for general functions it does not … Read more

New quasi-Newton method for solving systems of nonlinear equations

In this report, we propose the new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but it is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires $O(n^2)$ … Read more

Discrete Approximation of Two-Stage Stochastic and Distributionally Robust Linear Complementarity Problems

In this paper, we propose a discretization scheme for the two-stage stochastic linear complementarity problem (LCP) where the underlying random data are continuously distributed. Under some moderate conditions, we derive qualitative and quantitative convergence for the solutions obtained from solving the discretized two-stage stochastic LCP (SLCP). We ex- plain how the discretized two-stage SLCP may … Read more

K-Adaptability in Two-Stage Mixed-Integer Robust Optimization

We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying … Read more

Extended Formulations for Column Constrained Orbitopes

In the literature, packing and partitioning orbitopes were discussed to handle symmetries that act on variable matrices in certain binary programs. In this paper, we extend this concept by restrictions on the number of 1-entries in each column. We develop extended formulations of the resulting polytopes and present numerical results that show their effect on … Read more