Solving Lift-and-Project Relaxations of Binary Integer Programs

We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lov\’asz and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss … Read more

Necessary and Sufficient Optimality Conditions for Mathematical Programs with Equilibrium Constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient or locally sufficient for optimality under some MPEC … Read more

Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J.Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximity for linear optimization (LO) problems. They have also extended the approach for LO to SDO. In … Read more

On the Relationship Between Convergence Rates of Discrete and Continuous Dynamical Systems

Considering iterative sequences that arise when the approximate solution $x_k$ to a numerical problem is updated by $x_{k+1} = x_k+v(x_k)$, where $v$ is a vector field, we derive necessary and sufficient conditions for such discrete methods to converge to a stationary point of $v$ at different Q-rates in terms of the differential properties of $v$ … Read more