Polyhedral approximations in<i>p</i>-order cone programming

Vinel, Alexander; Krokhmal, Pavlo A.

doi:10.1080/10556788.2013.877905

Cited by 22 publications

(5 citation statements)

References 22 publications

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“…Note that it has been previously observed (see, Vielma et al (2008); Vinel and Krokhmal (2014b) and Chapter 3 of the current work) that ε 1 can be selected to be relatively large and still provide promising computational results, which explains the relation ε 2 ε 1 above. Note also that in this case, the described procedure can be viewed as repetitive resolving of a relatively small-scale LPs due to P 1 , guided by a regular branch-and-bound, with occasional calls to a large-scale P 2 .…”

Section: Branch-and-bound Methodssupporting

confidence: 65%

Mathematical programming techniques for solving stochastic optimization problems with certainty equivalent measures of risk

Vinel¹

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who provided invaluable feedback and advice that significantly improved this manuscript, as well as made the process of writing and defending much more enjoyable. It is also my pleasure to thank Maciej Rysz and Yana Morenko for fruitful collaborations. My great thanks go to my fellow students Dmitri Chernikov, Nathaniel Richmond and Bo Sun for helping to create and maintain a positive learning and research atmosphere. I especially appreciate the help of Mohammad Mirghorbani who was one of the first to welcome me in the new country after I moved to the United States.

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Section: Branch-and-bound Methodssupporting

confidence: 65%

Mathematical programming techniques for solving stochastic optimization problems with certainty equivalent measures of risk

Vinel¹

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“…Thus, in addition to the SOCP-based approaches for solving the pOCP problem (28) discussed earlier, we also employ an exact polyhedral-based approach with O −1 iteration complexity that was proposed in Vinel and Krokhmal (2014c). It consists in reformulating the p-order cone w 0 ≥ w 1 w N p via a set of three-dimensional p-cones…”

Section: Socp Reformulation Of P-ordermentioning

confidence: 99%

“…The process is initialized with j = 1 , 1 = /4, j = 1 N − 1, and continues until no violations of condition (35) are found. In Vinel and Krokhmal (2014c), it was shown that this cuttingplane procedure generates an -approximate solution to pOCP problem (28) within O −1 iterations.…”

Section: Socp Reformulation Of P-ordermentioning

confidence: 99%

“…An Exact Solution Method for pOCP Programs Based on Polyhedral Approximations. Computational methods for solving pOCP problems that are based on polyhedral approximations (Krokhmal andSoberanis 2010, Vinel andKrokhmal 2014c) represent an alternative to interior-point approaches, and can be beneficial in situations when a pOCP problem needs to be solved repeatedly, with small variations in problem data or problem structure.…”

Section: Socp Reformulation Of P-ordermentioning

confidence: 99%

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A Scenario Decomposition Algorithm for Stochastic Programming Problems with a Class of Downside Risk Measures

Rysz

Vinel

Krokhmal

et al. 2015

INFORMS Journal on Computing

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W e present an efficient scenario decomposition algorithm for solving large-scale convex stochastic programming problems that involve a particular class of downside risk measures. The considered risk functionals encompass coherent and convex measures of risk that can be represented as an infimal convolution of a convex certainty equivalent, and include well-known measures, such as conditional value-at-risk, as special cases. The resulting structure of the feasible set is then exploited via iterative solving of relaxed problems, and it is shown that the number of iterations is bounded by a parameter that depends on the problem size. The computational performance of the developed scenario decomposition method is illustrated on portfolio optimization problems involving two families of nonlinear measures of risk, the higher-moment coherent risk measures, and log-exponential convex risk measures. It is demonstrated that for large-scale nonlinear problems the proposed approach can provide up to an order-of-magnitude improvement in computational time in comparison to state-of-the-art solvers, such as CPLEX, Gurobi, and MOSEK.

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“…We mention an important way of constructing polyhedral approximations to the second-order cone is proposed by Ben-Tal and Nemirovski in [7], which exploits rotational symmetry. This approximation is first applied to MISOCP in [47] and further studied in [49].…”

Section: (Miqcp)mentioning

confidence: 99%

Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

Dong,

Luo

2018

Preprint

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Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by convex MIQCPs, which can be solved by MISOCP solvers. For nonconvex MIQCP with n variables and m general quadratic constraints, our method yields relaxations with at most O(n log(1/ε)) number of continuous/binary variables and linear constraints, together with m convex quadratic constraints, where ε is the approximation accuracy. The main novelty of our method lies in a very compact lifted mixed-integer formulation for approximating the (scalar) square function. This is derived by first embedding the square function into the boundary of a three-dimensional second-order cone, and then exploiting rotational symmetry in a similar way as in the construction of BenTal-Nemirovski approximation. We further show that this lifted formulation characterize the union of finite number of simple convex sets, which naturally relax the square function in a piecewise manner with properly placed knots. We implement (with JuMP) a simple adaptive refinement algorithm. Numerical experiments on synthetic instances used in the literature show that our prototypical implementation (with hundreds of lines of Julia code) can already close a significant portion of gap left by various state-of-the-art global solvers on more difficult instances, indicating strong promises of our proposed approach.

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Polyhedral approximations inp-order cone programming

Cited by 22 publications

References 22 publications

Mathematical programming techniques for solving stochastic optimization problems with certainty equivalent measures of risk

Mathematical programming techniques for solving stochastic optimization problems with certainty equivalent measures of risk

A Scenario Decomposition Algorithm for Stochastic Programming Problems with a Class of Downside Risk Measures

Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

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