An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

Nagarajan, Harsha; Lu, Mowen; Wang, Site; Bent, Russell; Sundar, Kaarthik

doi:10.1007/s10898-018-00734-1

Cited by 64 publications

(46 citation statements)

References 51 publications

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“…(9) are then derived through linear algebra on Eqs. (10) and (11). Similarly, it is easy to verify that the above point p * 1 ∈ X 1 .…”

Section: Disjunctive Formulation F λmentioning

confidence: 67%

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Convex hull formulations for mixed-integer multilinear functions

Nagarajan¹,

Sundar²,

Hijazi³

et al. 2019

AIP Conference Proceedings

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In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables. We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces. We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches.

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“…(9) are then derived through linear algebra on Eqs. (10) and (11). Similarly, it is easy to verify that the above point p * 1 ∈ X 1 .…”

Section: Disjunctive Formulation F λmentioning

confidence: 67%

“…(1) describes the convex hull of X B x (see [10]). For a multilinear function, ϕ x (x) = i∈I x i , standard global optimization methods apply recursive McCormick relaxations sequentially on bilinear terms, which do not necessarily capture the convex hull of the graph of ϕ x (x) [12,11].…”

Section: Given This Notation An Mimfmentioning

confidence: 99%

Convex hull formulations for mixed-integer multilinear functions

Nagarajan¹,

Sundar²,

Hijazi³

et al. 2019

AIP Conference Proceedings

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“…We implement an extensive testing infrastructure with hundreds of unit tests. Since Pajarito's first release, several other MINLP solvers have been written in Julia and are available through MathProgBase, such as POD [Nagarajan et al, 2017], Juniper [Kröger et al, 2018], and Katana. 16 Pajarito is integrated with the powerful MathProgBase abstraction layer.…”

Section: Julia and Mathprogbasementioning

confidence: 99%

Outer approximation with conic certificates for mixed-integer convex problems

2020

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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.

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“…The focus of this paper is a novel approach for globally optimizing the ACOPF. Our work is built on an adaptive multivariate partitioning algorithm (AMP) proposed in [9], [10]. The approach is based on a two-stage algorithm that uses sBB-like methods tailored to OPF problems.…”

Section: Introductionmentioning

confidence: 99%

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

Nagarajan

Bent

et al. 2018

2018 Power Systems Computation Conference (PSCC)

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Since the alternating current optimal power flow (ACOPF) problem was introduced in 1962, developing efficient solution algorithms for the problem has been an active field of research. In recent years, there has been increasing interest in convex relaxations-based solution approaches that are often tight in practice. Based on these approaches, we develop tight piecewise convex relaxations with convexhull representations, an adaptive, multivariate partitioning algorithm with bound tightening that progressively improves these relaxations and, given sufficient time, converges to the globally optimal solution. We illustrate the strengths of our algorithm using benchmark ACOPF test cases from the literature. Computational results show that our novel algorithm reduces the best-known optimality gaps for some hard ACOPF cases.

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An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

Cited by 64 publications

References 51 publications

Convex hull formulations for mixed-integer multilinear functions

Convex hull formulations for mixed-integer multilinear functions

Outer approximation with conic certificates for mixed-integer convex problems

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

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