Abstract. Let (M QP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (M QP ), i.e. we reformulate (M QP ) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of (M QP ). Computational experiences are carried out with instances of (M QP ) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in one hour of CPU time by a standard solver.
This article reports findings from a participative and qualitative study conducted with children who had experienced domestic violence, focusing on their perspectives on their relationships with their mothers. Three focus groups and 46 individual interviews were conducted with children to gather their experiences. The research findings demonstrate that women's and children's victimizations are inextricably linked, and that domestic violence affects mother-child relationships. They also show that, despite the challenges and difficulties, children generally consider their mothers as very significant individuals in their lives, and have close relationships with them. The findings also reveal a dynamic of mutual protectiveness.
We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.
The maximum-entropy sampling problem is a fundamental and challenging combinatorial-optimization problem, with application in spatial statistics. It asks to find a maximum-determinant order-s principal submatrix of an order-n covariance matrix. Exact solution methods for this NP-hard problem are based on a branch-and-bound framework. Many of the known upper bounds for the optimal value are based on convex optimization. We present a methodology for "mixing" these bounds to achieve better bounds.
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