A gauge function f(.) is a nonnegative convex function that is positively homogeneous and satisfies f(O)=O. Norms and pseudonorms are specific instances of a gauge function. This paper presents a gauge duality theory for a gauge program, which is the problem of minimizing the value of a gauge function f(.) over a convex set. The gauge dual program is also a gauge program, unlike the standard La...

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algo...

In this paper we consider a fractional optimization problem that minimizes the ratio of two quadratic functions subject to a strictly convex quadratic constraint. First using the extension of Charnes-Cooper transformation, an equivalent homogenized quadratic reformulation of the problem is given. Then we show that under certain assumptions, it can be solved to global optimality using semidefini...

Let F be a compact subset of the n-dimensional Euclidean space Rn represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets Ck (k = 1, 2, . . . ) of Rn such that (...

We present a new tree-search algorithm for solving mixed-integer nonlinear programs (MINLPs). Rather than relying on computationally expensive nonlinear solves at every node of the branchand-bound tree, our algorithm solves a quadratic approximation at every node. We show that the resulting algorithm retains global convergence properties for convex MINLPs, and we present numerical results on a ...

We investigate the computational potential of split inequalities for non-convex quadratic integer programming, first introduced by Letchford [11] and further examined by Burer and Letchford [8]. These inequalities can be separated by solving convex quadratic integer minimization problems. For small instances with box-constraints, we show that the resulting dual bounds are very tight; they can c...

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems...

Bounds on the optimal value of a convex 0-1 quadratic programming problem with linear constraints can be improved by a preprocessing step that adds to the quadratic objective function terms which are equal to 0 for all 0-1 feasible solutions yet increase its continuous minimum. The continuous and the CHR bounds are improved if one first uses Plateau’s QCR method (2005), or one of its predecesso...

Bounds on the optimal value of a convex 0-1 quadratic programming problem with linear constraints can be improved by a preprocessing step that adds to the quadratic objective function terms which are equal to 0 for all 0-1 feasible solutions yet increase its continuous minimum. The continuous and the CHR bounds are improved if one first uses Plateau’s QCR method (2005), or one of its predecesso...

In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength being restricted to be i...