In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance … Read more
In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that … Read more
In a recent paper, Dash, Dey and Gunluk (2010) showed that many families of inequalities for the two-row continuous group relaxation and variants of this relaxation are cross cuts or crooked cross cuts, both of which generalize split cuts. Li and Richard (2008) recently studied t-branch split cuts for mixed-integer programs for integers t >= … Read more
We consider a general class of second-order iterations for unconstrained optimization that includes regularization and trust-region variants of Newton’s method. For each method in this class, we exhibit a smooth, bounded-below objective function, whose gradient is globally Lipschitz continuous within an open convex set containing any iterates encountered and whose Hessian is $\alpha-$Holder continuous (for … Read more
Let $K \subset \mathbb R^n$ be a regular convex cone, let $e_1,\dots,e_n \in \partial K$ be linearly independent points on the boundary of a compact affine section of the cone, and let $x^* \in K^o$ be a point in the relative interior of this section. For $k = 1,\dots,n$, let $l_k$ be the line through … Read more
We consider bilevel pseudomonotone equilibrium problems. We use a penalty function to convert a bilevel problem into one-level ones. We generalize a pseudo $\nabla$-monotonicity concept from $\nabla$-monotonicity and prove that under pseudo $\nabla$-monotonicity property any stationary point of a regularized gap function is a solution of the penalized equilibrium problem. As an application, we discuss … Read more
We formulate minmax flow problems as a DC. optimization problem. We then apply a DC primal-dual algorithm to solve the resulting problem.The obtained computational results show that the proposed algorithm is efficient thanks to particular structures of the minmax flow problems. Citation 1. An L. T. H. and Tao P. D., The DC (Difference of … Read more
Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality \begin{equation} \label{le} per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n \end{equation} We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\ For all … Read more
We present a constraint-and-column generation algorithm to solve two-stage robust optimization problems. Compared with existing Benders style cutting plane methods, it is a general procedure with a unified approach to deal with optimality and feasibility. A computational study on a two-stage robust location-transportation problem shows that it performs an order of magnitude faster. Also, it … Read more
The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is $NP$-hard and a popular approach replaces the rank function with the nuclear norm of the … Read more