We show, using elementary considerations, that a modified barrier function method for the solution of convex programming problems converges for any fixed positive setting of the barrier parameter. With mild conditions on the primal and dual feasible regions, we show how to use the modified barrier function method to obtain primal and dual optimal solutions, even in the presence of degeneracy. W...

Non-Convex Quadratic Programming with Box Constraints is a fundamental NP-hard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and vertices, show their invariance under certain affine...

This chapter surveys key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.

In this paper we present an extremely general method for approximately solving a large family of convex programs where the solution can be divided between different agents, subject to joint differential privacy. This class includes multi-commodity flow problems, general allocation problems, and multi-dimensional knapsack problems, among other examples. The accuracy of our algorithm depends on t...

This paper presents the robust optimization framework in the modeling language YALMIP, which carries out robust modeling and uncertainty elimination automatically, and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modeling and tractability of complex convex uncertain op...

Symmetry is the essential element of lifted inference that has recently demonstrated the possibility to perform very efficient inference in highly-connected, but symmetric probabilistic models models. This raises the question, whether this holds for optimisation problems in general. Here we show that for a large class of optimisation methods this is actually the case. More precisely, we introdu...

In this paper, a recurrent neural network model is proposed for solving non-smooth convex programming problems, which is a natural extension of the previous neural networks. By using the non-smooth analysis and the theory of differential inclusions, the global convergence of the equilibrium is analyzed and proved. One simulation example shows the convergence of the presented neural network.

Most research in robust optimization has so far been focused on inequality-only, convex conic programming with simple linear models for uncertain parameters. Many practical optimization problems, however, are nonlinear and non-convex. Even in linear programming, coefficients may still be nonlinear functions of uncertain parameters. In this paper, we propose robust formulations (see (1) versus (...

This work presents a non-convex variational approach to joint image reconstruction and labeling. Our regularization strategy, based on the KL-divergence, takes into account the smooth geometry on the space of discrete probability distributions. The proposed objective function is efficiently minimized via DC programming which amounts to solving a sequence of convex programs, with guaranteed conv...

We investigate the computational potential of split inequalities for non-convex quadratic integer programming, first introduced by Letchford [11] and further examined by Burer and Letchford [8]. These inequalities can be separated by solving convex quadratic integer minimization problems. For small instances with box-constraints, we show that the resulting dual bounds are very tight; they can c...