The uniform bound of 1-norm is given for the inverse of lower triangular Toeplitz matrices with nonnegative monotonic decreasing entries whose limit is zero. The new bound is the sharpest under the given constraints. This result is then employed to resolve a long standing open problem posed by Brunner concerning the convergence of the one-point … Read more
We investigate fast methods that allow to quickly eliminate variables (features) in supervised learning problems involving a convex loss function and a l1 -norm penalty, leading to a potentially substantial reduction in the number of variables prior to running the supervised learning algorithm. The methods are not heuristic: they only eliminate features that are guaranteed … Read more
The proximal point algorithm (PPA) is classical and popular in the community of Optimization. In practice, inexact PPAs which solves the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact PPA with a new inexact criterion for solving convex minimization, and show that the … Read more
The cutting plane tree (CPT) algorithm provides a finite disjunctive programming procedure to obtain the solution of general mixed-integer linear programs (MILP) with bounded integer variables. In this paper, we present our computational experience with variants of the CPT algorithm. Because the CPT algorithm is based on discovering multi-term disjunctions, this paper is the first … Read more
This article extends a previous work focused on a mixed integer programming (MIP) based heuristic approach, aimed at solving non-standard three-dimensional problems with additional conditions. The paper that follows considers a mixed integer non-linear (MINLP) reformulation of the previous model, to improve the former heuristic, based on linear relaxation. The approach described herewith is addressed, … Read more
In their paper “Duality of linear conic problems” A. Shapiro and A. Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is “If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)”, while property (B) is “If both … Read more
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension of some useful techniques from these two fields to a … Read more
This paper proposes a framework for trade-off analyses of blackbox constrained optimization problems. Two strategies are developed to show the trade-off of the optimal objective function value with tightening or loosening general constraints. These are a simple method which may be performed immediately after a single optimization and a detailed method performing biobjective optimization on … Read more
Geometric random walks have been proposed and analyzed for solving optimization problems. In this paper we report our computational experience with generating feasible integer solutions of mixed-integer linear programs using geometric random walks, and a random ray approach. A feasibility pump is used to heuristically round the generated points. Computational results on MIPLIB2003 and COR@L … Read more
We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound … Read more