arXiv subjects

Chris Coey

Publications and source records attributed to Chris Coey.

5 recordsLinked to original sources

arXiv2022

Performance enhancements for a generic conic interior point algorithm

In recent work, we provide computational arguments for expanding the class of proper cones recognized by conic optimization solvers, to permit simpler, smaller, more natural conic formulations. We define an exotic cone as a proper cone for which we can implement a small set of tractable (i.e. fast, numerically stable, analytic) oracles for a logarithmically homogeneous self-concordant barrier for the cone or for its dual cone. Our extensible, open source conic interior point solver, Hypatia, allows modeling and solving any conic optimization problem over a Cartesian product of exotic cones. In this paper, we introduce Hypatia's interior point algorithm. Our algorithm is based on that of Skajaa and Ye [2015], which we generalize by handling exotic cones without tractable primal oracles. With the goal of improving iteration count and solve time in practice, we propose a sequence of four enhancements to the interior point stepping procedure of Skajaa and Ye [2015]: (1) loosening the central path proximity condition, (2) adjusting the directions using a third order directional derivative barrier oracle, (3) performing a backtracking search on a curve, and (4) combining the prediction and centering directions. We implement 23 useful exotic cones in Hypatia. We summarize the complexity of computing oracles for these cones, showing that our new third order oracle is not a bottleneck, and we derive efficient and numerically stable oracle implementations for several cones. We generate a diverse benchmark set of 379 conic problems from 37 different applied examples. Our computational testing shows that each stepping enhancement improves Hypatia's iteration count and solve time. Altogether, the enhancements reduce the shifted geometric means of iteration count and solve time by over 80% and 70% respectively.

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arXiv2022

Sum of squares generalizations for conic sets

In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and Yildiz [18], using the sum of squares cone directly in a nonsymmetric interior point algorithm. Beyond nonnegativity, more complicated polynomial constraints (in particular, generalizations of the positive semidefinite, second order and $\ell_1$-norm cones) can also be modeled through structured sum of squares programs. We take a different approach and propose using more specialized polynomial cones instead. This can result in lower dimensional formulations, more efficient oracles for interior point methods, or self-concordant barriers with smaller parameters. In most cases, these algorithmic advantages also translate to faster solving times in practice.

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arXiv2021

Conic optimization with spectral functions on Euclidean Jordan algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, i.e. proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones - the root-determinant cones and the matrix monotone derivative cones - we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.

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arXiv2021

Solving natural conic formulations with Hypatia.jl

Many convex optimization problems can be represented through conic extended formulations with auxiliary variables and constraints using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. Such extended formulations are often significantly larger and more complex than equivalent conic natural formulations, which can use a much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement tractable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. In this paper we introduce Hypatia, a highly-configurable open-source conic primal-dual interior point solver with a generic interface for exotic cones. Hypatia is written in Julia and accessible through JuMP, and currently implements around two dozen useful predefined cones (some with multiple variants). We define some of Hypatia's exotic cones, and for conic constraints over these cones, we analyze techniques for constructing equivalent representations using the standard cones. For optimization problems from a variety of applications, we introduce natural formulations using these exotic cones, and we show that the natural formulations are simpler and lower-dimensional than the equivalent extended formulations. Our computational experiments demonstrate the potential advantages, especially in terms of solve time and memory usage, of solving the natural formulations with Hypatia compared to solving the extended formulations with either Hypatia or MOSEK 9.

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arXiv2018

Outer Approximation With Conic Certificates For Mixed-Integer Convex Problems

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 $\mathcal{K}^*$ cuts derived from conic certificates for continuous primal-dual 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 $\mathcal{K}^*$ 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 $\mathcal{K}^*$ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful $\mathcal{K}^*$ cuts. Our new open source MI-conic solver Pajarito (http://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 $\mathcal{K}^*$ cuts.

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