The Legendre-Fenchel Conjugate of the Product of Two positive definite Quadratic Forms

It is well-known that the Legendre-Fenchel conjugate of a positive definite quadratic form can be explicitly expressed as another positive definite quadratic form, and that the conjugate of the sum of several positive definite quadratic forms can be expressed via inf-convolution. However, the Legendre-Fenchel conjugate of the product of two positive definite quadratic forms is … Read more

Necessary Optimality Conditions for two-stage Stochastic Programs with Equilibrium Constraints

Developing first order optimality conditions for a two-stage stochastic mathematical program with equilibrium constraints (SMPEC) whose second stage problem has multiple equilibria/solutions is a challenging undone work. In this paper we take this challenge by considering a general class of two-stage whose equilibrium constraints are represented by a parametric variational inequality (where the first stage … Read more

Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in: [Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization, 2,71–109, 1998.] Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP … Read more

Hager-Zhang Active Set Algorithm for Large-Scale Continuous Knapsack Problems

The structure of many real-world optimization problems includes minimization of a nonlinear (or quadratic) functional subject to bound and singly linear constraints (in the form of either equality or bilateral inequality) which are commonly called as continuous knapsack problems. Since there are efficient methods to solve large-scale bound constrained nonlinear programs, it is desirable to … Read more

Sparse Signal Reconstruction via Iterative Support Detection

We present a novel sparse signal reconstruction method “ISD”, aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical l_1 minimization approach. ISD addresses failed reconstructions of l_1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new … Read more

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions

In this paper, we apply the quadratic penalization technique to derive strong Lagrangian duality property for an inequality constrained invex program. Our results extend and improve the corresponding results in the literature. Citation Bazara, M. S. and Shetty, C. M., Nonlinear Programming Theory and Algorithms, John Wiley \& Sons, New York, 1979. Ben-Israel, A. and … Read more

Block Structured Quadratic Programming for the Direct Multiple Shooting Method for Optimal Control

In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved … Read more

A Comparison of Lower Bounds for the Symmetric Circulant Traveling Salesman Problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare … Read more

Perspective Reformulation and Applications

In this paper we survey recent work on the perspective reformulation approach that generates tight, tractable relaxations for convex mixed integer nonlinear programs (MINLP)s. This preprocessing technique is applicable to cases where the MINLP contains binary indicator variables that force continuous decision variables to take the value 0, or to belong to a convex set. … Read more

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully-adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right hand side of the constraints is … Read more