Extended Farkas lemma and strong duality for composite optimization problems with DC functions

In this paper, we consider the optimization problem in locally convex Hausdorff topological vector spaces with objectives given as the difference of two composite functions and constraints described by an arbitrary (possibly infinite) number of convex inequalities. Using the epigraph technique, we introduce some new constraint qualifications, which completely characterize the Farkas lemma, the dualities between the primal problem and its dual problem. Applications to the conical programming with DC composite function are also given.     

New optimality conditions for quadratic optimization problems with binary constraints

In this article, we obtain new sufficient optimality conditions for the nonconvex quadratic optimization problems with binary constraints by exploring local optimality conditions. The relation between the optimal solution of the problem and that of its continuous relaxation is further extended.     

A robust algorithm for quadratic optimization under quadratic constraints

Most existing methods of quadratically constrained quadratic optimization actually solve a refined linear or convex relaxation of the original problem. It turned out, however, that such an approach may sometimes provide an infeasible solution which cannot be accepted as an approximate optimal solution in any reasonable sense. To overcome these limitations a new approach is proposed that guarantees a more appropriate approximate optimal solution which is also stable under small perturbations of the constraints.     

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.     

A robust secant method for optimization problems with inequality constraints

This paper presents a secant method, based on R. B. Wilson's formula for the solution of optimization problems with inequality constraints. Global convergence properties are ensured by grafting the secant method onto a phase I - phase II feasible directions method, using a rate of convergence test for crossover control.This research was sponsored by the National Science Foundation, Grant No. ENG-73-08214 and Grant No. (RANN)-ENV-76-04264, and by the Joint Services Electronics Program. Contract No. F44620-76-C-0100.