Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems These problems are in general NP hard and numerically intractable Lagrangian relaxations often provide good approximate solutions to these hard problems Such relaxations are equivalent to semidefinite programming relaxations
For several special cases of QQP, eg, convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, ie, there is a zero duality gap However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective 
In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT=I For this problem we show that the Lagrangian dual based on relaxing the constraints XXT=I and the seemingly redundant constraints XT X=I has a zero duality gap This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the max-cut problem

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On Lagrangian Relaxation of Quadratic Matrix Constraints

We generalize many of the results on efficient points for linear multiple objective optimization problems to the nonlinear case by focusing on an auxiliary problem. The approach, which relies on duality theory, is a straightforward development that even in the linear case yields simpler proofs.

Efficiency in multiple objective optimization problems

Linear Programming: Methods and Applications

Let f0, f1, …, fm be convex functions on Rn, and let (P) denote the problem of minimizing f0(x) subject to f1(x) ≤ 0, …, fm(x) ≤ 0. According to the theory of conjugate functions, many different dual problems can be associated with (P), each one corresponding to a particular class of perturbations of (P). Thus, in developing dual methods of solutions of (P), one has considerable flexibility in the choice of the dual problem, and the choice can be made in view of its suitability for a given purpose. This paper treats some simple possibilities in the important case where each of the functions fi satisfies the following condition: fi is not affine along any line segment, unless it is affine along the entire line extending the segment (The latter holds, for example, if fi is analytic.) It is shown that the perturbations can be chosen in this case sothat the corresponding dual problem (P*) consists essentially of maximizing a differentiable concave function subject to linear constraints. The duality theorems applicable to (P) and(P*) are then somewhat more refined than those in the general theory; e.g. the infimum in (P) and the supremum in (P*) are necessarily equal, if (P) is consistent. The duality theory for the geometric programs of Duffin, Peterson and Zener, and the quadratic and lp programs of Peterson and Ecker, is derived as an illustration.

Elmor L. Peterson

Papers

Geometric programming: Duality in quadratic programming and lp-approximation III (degenerate programs)

An economic interpretation of duality in linear programming