The primal-dual hybrid gradient method reduces to a primal method for linearly constrained optimization problems

In this work, we show that for linearly constrained optimization problems the primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can be written as an entirely primal algorithm. This allows us to prove convergence of the iterates even in the degenerate cases when the linear system is inconsistent or when the strong duality … Read more

Local convergence analysis of the Levenberg-Marquardt framework for nonzero-residue nonlinear least-squares problems under an error bound condition

The Levenberg-Marquardt method (LM) is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares prob- lems. In this paper, we consider local convergence issues of the LM method when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full-rank of the Jacobian in a … Read more

On Lifted Cover Inequalities: A New Lifting Procedure with Unusual Properties

Lifted cover inequalities are well-known cutting planes for 0-1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield facet-defining … Read more

An Integer Programming Formulation of the Key Management Problem in Wireless Sensor Networks

With the advent of modern communications systems, much attention has been put on developing methods for securely transferring information between constituents of wireless sensor networks. To this effect, we introduce a mathematical programming formulation for the key management problem, which broadly serves as a mechanism for encrypting communications. In particular, an integer programming model of … Read more

Adaptive Cubic Regularization methods with dynamic inexact Hessian information and applications to finite-sum minimization

We consider the Adaptive Regularization with Cubics approach for solving nonconvex optimization problems and propose a new variant based on inexact Hessian information chosen dynamically. The theoretical analysis of the proposed procedure is given. The key property of ARC framework, constituted by optimal worst-case function/derivative evaluation bounds for first- and second-order critical point, is guaranteed. … Read more

Improved Decision Rule Approximations for Multi-Stage Robust Optimization via Copositive Programming

We study decision rule approximations for generic multi-stage robust linear optimization problems. We consider linear decision rules for the case when the objective coefficients, the recourse matrices, and the right-hand sides are uncertain, and consider quadratic decision rules for the case when only the right-hand sides are uncertain. The resulting optimization problems are NP-hard but … Read more

Outer Approximation With Conic Certificates For Mixed-Integer Convex Problems

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K* cuts} derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all … Read more

Bilevel optimization: theory, algorithms and applications

Bilevel optimization problems are hierarchical optimization problems where the feasible region of the so-called upper level problem is restricted by the graph of the solution set mapping of the lower level problem. Aim of this article is to collect a large number of references on this topic, to show the diversity of contributions and to … Read more

Asynchronous Sequential Inertial Iterations for Common Fixed Points Problems with an Application to Linear Systems

The common fixed points problem requires finding a point in the intersection of fixed points sets of a finite collection of operators. Quickly solving problems of this sort is of great practical importance for engineering and scientific tasks (e.g., for computed tomography). Iterative methods for solving these problems often employ a Krasnosel’skii-Mann type iteration. We … Read more

A fundamental proof to convergence analysis of alternating direction method of multipliers for weakly convex optimization

The convergence analysis of the alternating direction method of multipliers (ADMM) methods to convex/nonconvex combinational optimization have been well established in literature. Consider the extensive applications of weakly convex function in signal processing and machine learning(e.g. \textit{Special issue: DC-Theory, Algorithms and Applications, Mathematical Programming, 169(1):1-336,2018}), in this paper, we investigate the convergence analysis of ADMM … Read more