An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs

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

doi:10.48550/arxiv.1707.02514

Cited by 6 publications

(13 citation statements)

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“…These resemble SOS-II constraints. The formulation in (10) has many interesting polyhedral properties. For example, the projection of this polytope on to the space of {x1, x2, x3, x} has integral extreme points.…”

Section: Piecewise Trilinear Functionsmentioning

confidence: 99%

“…To globally solve the ACOPF problem, we use the Adaptive Multivariate Partitioning (AMP) algorithm described in [9], [10]. The key idea of AMP is that AMP leverages the observations that solutions based on relaxations to ACOPF are often tight in practice and that locally optimal solutions are also very good [4].…”

Section: Adaptive Multivariate Partitioningmentioning

confidence: 99%

“…Line 5 describes our variable selection strategy for partitioning (discussed later). This is the main point of departure (and contribution) from the AMP algorithm discussed in [10]. Line 6 initializes the piecewise partitions of variables in X around those variables' assignments in σ (see Figure 4).…”

Section: Adaptive Multivariate Partitioningmentioning

confidence: 99%

“…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%

“…This approach exploits the observation that local solutions to standard benchmark instances are already very good. Our recent results on generic mixed-integer nonlinear programs suggest that refining variable domains adaptively around best-known feasible solutions can dramatically speed up the convergence to global optimum [10], [13].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Piecewise Trilinear Functionsmentioning

confidence: 99%

Section: Adaptive Multivariate Partitioningmentioning

confidence: 99%

Section: Adaptive Multivariate Partitioningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia

Kröger

Coffrin

Hijazi

et al. 2018

Integration of Constraint Programming, Artificial Intelligence, and Operations Research

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Nonconvex mixed-integer nonlinear programs (MINLPs) represent a challenging class of optimization problems that often arise in engineering and scientific applications. Because of nonconvexities, these programs are typically solved with global optimization algorithms, which have limited scalability. However, nonlinear branch-and-bound has recently been shown to be an effective heuristic for quickly finding highquality solutions to large-scale nonconvex MINLPs, such as those arising in infrastructure network optimization. This work proposes Juniper, a Julia-based open-source solver for nonlinear branch-and-bound. Leveraging the high-level Julia programming language makes it easy to modify Juniper's algorithm and explore extensions, such as branching heuristics, feasibility pumps, and parallelization. Detailed numerical experiments demonstrate that the initial release of Juniper is comparable with other nonlinear branch-and-bound solvers, such as Bonmin, Minotaur, and Knitro, illustrating that Juniper provides a strong foundation for further exploration in utilizing nonlinear branch-and-bound algorithms as heuristics for nonconvex MINLPs.

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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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An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs

Cited by 6 publications

References 0 publications

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia

Outer approximation with conic certificates for mixed-integer convex problems

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