We consider the non-convex quadratic maximization problem subject to the l1 unit ball constraint. The nature of the l1 norm structure makes this problem extremely hard to analyze, and as a consequence, the same difficulties are encountered when trying to build suitable approximations for this problem by some tractable convex counterpart formulations. We explore some properties of this problem, derive SDP-like relaxations and raise open questions.

The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated.

In this paper, we have studied the problem of minimizing the ratio of two indefinite quadratic functions subject to a strictly convex quadratic constraint. First utilizing the relationship between fractional and parametric programming problems due to Dinkelbach, we reformulate the fractional problem as a univariate equation. To find the root of the univariate equation, the generalized Newton method is utilized that requires solving a nonconvex quadratic optimization problem at each iteration. A...

We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter...

$(max,+)$-linear functions are functions which can be expressed as the maximum of a finite number of linear functions of one variable having the form $f(x_1,\cdots ,x_h)={max}_j(a_j+x_j)$, where $a_j$, $j=1,\cdots ,h$, are real numbers. Similarly $(min,+)$-linear functions are defined. We will consider optimization problems in which the set of feasible solutions is the solution set of a finite inequality system, where the inequalities have $(max,+)$-linear functions of variables $x$ on one side and $(min,+)$-linear functions of variables $y$ on the other side. Such systems can be applied...

Many algorithms for globally solving sum of affine ratios problem (SAR) are based on equivalent problem and branch-and-bound framework. Since the exhaustiveness of branching rule leads to a significant increase in the computational burden for solving the equivalent problem. In this study, a new range reduction method for outcome space of the denominator is presented for globally solving the sum of affine ratios problem (SAR). The proposed range reduction method offers a possibility to delete a large...