Differential properties of Euclidean projection onto power cone

Hien, Le Thi Khanh

doi:10.1007/s00186-015-0514-0

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…In theory, R. Chares [4] proposes two important concepts (i.e., α-representable and extended α-representable, see Appendix 6.1) involving powers and exponentials and plenty of famous cones can be generated from these two cones such as second-order cones [1,22,14,7,9,24], p-order cones [2,27,44], geometric cones [3,15,16,26], L p cones [17] and etc., one can refer to [4, chapter 4] for more examples. In applications, many practical problems can be cast into optimization models involving the power cone constraints and the exponential cone constraints, such as location problems [4,19] and geometric programming problems [3,31,34] (see Appendix 6.2). Therefore, it becomes quite obvious that there is a great demand for providing systematic studies for these cones.…”

Section: Motivations and Literaturesmentioning

confidence: 99%

The decompositions with respect to two core non-symmetric cones

Yang

Chen

et al. 2019

J Glob Optim

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It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones.However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conicconvexity, conic-monotonicity and etc. In this paper, motivated by Chares Robert's thesis [Chares, R.: Cones and interior-point algorithms for structured convex optimization involving powers and exponentials. PhD thesis, UCL-Universite Catholique de Louvain ( 2009)], we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart.Keywords Moreau decomposition theorem • power cone • exponential cone • non-symmetric cones.Mathematics Subject Classification (2000) 49M27 • 90C25. IntroductionConsider the following two core non-symmetric cones

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Section: Motivations and Literaturesmentioning

confidence: 99%

The decompositions with respect to two core non-symmetric cones

Yang

Chen

et al. 2019

J Glob Optim

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“…While this conic representation of the geometric mean is known in the literature [7], it is arguably unnecessarily complex for modelers to understand, and CVX, for example, provides a geo mean atom which transparently handles this transformation. Subsequent to the second-order and positive semidefinite cones, researchers have investigated the exponential cone [39], EXP = cl{(x, y, z) ∈ R 3 : y exp(x/y) ≤ z, y > 0}, (15) and the power cone [27],…”

Section: Midcp and Conic Representabilitymentioning

confidence: 99%

Polyhedral approximation in mixed-integer convex optimization

et al. 2017

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Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro

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“…The question of which functions can be represented by second-order cones has been well studied [23,6]. More recently, a number of authors have considered nonsymmetric cones, in particular the exponential cone, which can be used to model logarithms, entropy, logistic regression, and geometric programming [27], and the power cone, which can be used to model p-norms and powers [19]. Not representable Total 217 107 7 2 0 333 Table 1.…”

Section: Extended Formulations and Conic Representabilitymentioning

confidence: 99%

Extended Formulations in Mixed-Integer Convex Programming

Lubin

Yamangil

Bent

et al. 2016

Integer Programming and Combinatorial Optimization

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Abstract. We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.

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Differential properties of Euclidean projection onto power cone

Cited by 8 publications

References 18 publications

The decompositions with respect to two core non-symmetric cones

The decompositions with respect to two core non-symmetric cones

Polyhedral approximation in mixed-integer convex optimization

Extended Formulations in Mixed-Integer Convex Programming

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