Subgradient Based Outer Approximation for Mixed Integer Second Order Cone Programming

Symmetry-exploiting cuts for a class of mixed-0/1 second-order cone programs

“…For the sake of completeness we now briefly recall the concept of an outer approximation algorithm as proposed for example by [5], [11], [12], [18] or [14]. While executing such an algorithm iteratively, integer feasible solutions of a linear outer approximation problem are computed and nonlinear subproblems are solved that arise from the given mixed-integer nonlinear problem (here: (WSOC)) by fixing the integral part of the variables.…”

“…These two properties ensure that the outer approximation algorithm converges and every (potential) integral assignment is visited at most once. For details of the general case we refer to [12] and [11].…”

“…[18]. In that case, the Lagrange multipliers can be identified using the dual solutions of these SOCP representations, for details compare, e.g., [11].…”

“…It uses a branch-and-cut approach to solve the strong reformulation (SQE) as a mixed integer second order cone program. The second algorithm (Classic) uses a linear outer approximation method based on standard subgradient based linearizations (4.3) of the original nonlinear constraints in (WSOC), for details compare [11]. The third method (BSC) is an outer approximation method using the BSC-cuts (BSC-O) and (BSC-F) presented in this paper.…”

“…Due to their structure and convexity, the underlying relaxations can be solved in polynomial time to ε-optimality. Mixed-integer second-order cone programs are usually solved via outer approximation algorithms or within a branchand-cut framework (see e.g., [22], [11] or [6] for an overview).…”