We study the problem of decomposing the Hessian matrix of a Mixed-Integer Convex Quadratic Program into the sum of positive semidefinite 2×2 matrices. Solving this problem enables the use of Perspective Reformulation techniques for obtaining strong lower bounds for MICQPs with semi-continuous variables but a non-separable objective function. An explicit formula is derived for constructing 2×2 decompositions when the underlying matrix is Weakly Scaled Diagonally Dominant, and necessary and sufficient conditions are given for the decomposition to be unique. For matrices lying outside this class, two exact SDP approaches and an efficient heuristic are developed for finding approximate decompositions. We present preliminary results on the bound strength of a 2×2 Perspective Reformulation for the Portfolio Optimization Problem, showing that for some classes of instances the use of 2×2 matrices can significantly improve the quality of the bound w.r.t. the best previously known approach, although at a possibly high computational cost.

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This article considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP has been developed to solve this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is computed using the variable fixing method or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the dual function and computing the solution of the dual problem. Also, a new version of the variable fixing algorithm is developed that is convergent even when the objective Hessian is not strictly positive definite. The hybrid algorithm NAPHEAP that uses a Newton-type method (variable fixing method, secant method, or Newton's method) to bracket a root, followed by a heap-based monotone break point search, can be faster than a Newton-type method by itself, as demonstrated in the numerical experiments.

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