Outer approximation with conic certificates for mixed-integer convex problems

Abstract

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K * cuts derived from conic certificates for continuous primaldual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K * cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate K * cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K * cuts. Our new open source MI-conic solver Pajarito (github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear (MILP) solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX's specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K * cuts. * M. Lubin is now at Google. The authors thank MOSEK for access to alpha releases of MOSEK 9 and Russell Bent and Emre Yamangil for support for and early contributions to Pajarito.

A review and comparison of solvers for convex MINLP

“…Pavito is an open-source solver for convex MINLP implemented in Julia by C. Coey, M. Lubin and J. P. Vielma. Its functionality was previously part of the Pajarito solver for conic MINLP, but the NLP functionality was recently moved into the Pavito solver (Coey et al 2018). Contrary to the other solvers presented in this manuscript,…”

Feedback control of linear systems with optimal sensor and actuator selection

“…In this section, we consider two LTI case studies with one of them already discussed in Section 4 to compare the MBSDP-BNB approach with three other approaches, namely, IRLA (Singh et al, 2018b), MBSDP-OA (Coey et al, 2018), and finally with an ES.…”

“…For combined control design using MBSDP-OA, we use the Pajarito solver formulated in JuMP Dunning et al (2017) in Julia programming language and use Gurobi Optimization Inc (2014) and MOSEK as the subsolvers. Unlike YALMIP’s BNB, which relaxes the integer constraints during the iterative BNB algorithm, MBSDP-OA using Pajarito solver (Coey et al, 2018) relaxes the SDP constraints. The underlying idea is to convert the SDP constraints into a finite-dimensional linear problem with linear constraints which are called cutting planes obtaining outer approximation of the original SDP problem.…”

“…In Section 4, the mentioned approach is applied to seismic vibration control of a 20-story building for LTI controller synthesis and active vibration control of a composite plate for an LPV controller synthesis. In Section 5, a comparison study is carried out between different approaches available that are iterative reweighted ℓ 1 -norm algorithm (IRLA) (Singh et al, 2018b), outer approximations (MBSDP-OA) (Coey et al, 2018), and exhaustive search (ES). Discussions on comparison of methods and limitations are also carried out in this section.…”

A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids

“…The optimal solution of an integer-convex optimization problem can be guaranteed via the branch and bound method [21]. For each binary combination of variables, the internal problem is convex, implying that each internal problem has a global solution [22].…”