Copositive programming for mixed-binary quadratic optimization via Ising solvers

Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing … Read more

Duality assertions in vector optimization w.r.t. relatively solid convex cones in real linear spaces

We derive duality assertions for vector optimization problems in real linear spaces based on a scalarization using recent results concerning the concept of relative solidness for convex cones (i.e., convex cones with nonempty intrinsic cores). In our paper, we consider an abstract vector optimization problem with generalized inequality constraints and investigate Lagrangian type duality assertions … Read more

Duality aspects in convex conic programming

In this paper we study strong duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove strong alternatives to the existence of the relative interior point in the primal (dual) feasible set. We analyze the … Read more

Exact computation of an error bound for a generalized linear complementarity problem with unique solution

This paper considers a generalized form of the standard linear complementarity problem with unique solution and provides a more precise expression of an upper error bound discovered by Chen and Xiang in 2006. This expression has at least two advantages. It makes possible the exact computation of the error bound factor and it provides a … Read more

An Exact Projection-Based Algorithm for Bilevel Mixed-Integer Problems with Nonlinearities

We propose an exact global solution method for bilevel mixed-integer optimization problems with lower-level integer variables and including nonlinear terms such as, \eg, products of upper-level and lower-level variables. Problems of this type are extremely challenging as a single-level reformulation suitable for off-the-shelf solvers is not available in general. In order to solve these problems … Read more

Douglas-Rachford Splitting for Pathological Convex Optimization

Despite the vast literature on DRS, there has been very little work analyzing their behavior under pathologies. Most analyses assume a primal solution exists, a dual solution exists, and strong duality holds. When these assumptions are not met, i.e., under pathologies, the theory often breaks down and the empirical performance may degrade significantly. In this … Read more

Strong Duality: Without Simplex and without theorems of alternatives

The simplex method has its own problems related to degenerate basic feasible solutions. While such solutions are infrequent, from a theoretical standpoint a proof of the strong duality theorem that uses the simplex method is not complete until it has taken a few extra steps. Further, for economists the duality theorem is extremely important whereas … Read more

Sensitivity analysis of semidefinite programs without strong duality

Suppose that we are given a feasible conic program with a finite optimal value and with strong duality failing. It is known that there are small perturbations of the problem data that lead to relatively big changes in the optimal value. We quantify the notion of big change in the case of a semidefinite program … Read more

Trust-Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having … Read more

On the Augmented Lagrangian Dual for Integer Programming

We consider the augmented Lagrangian dual for integer programming, and provide a primal characterization of the resulting bound. As a corollary, we obtain proof that the augmented Lagrangian is a strong dual for integer programming. We are able to show that the penalty parameter applied to the augmented Lagrangian term may be placed at a … Read more