On p-norm linear discrimination

Morenko, Yana; Vinel, Alexander; Yu, Zhaohan; Krokhmal, Pavlo A.

doi:10.1016/j.ejor.2013.06.053

Cited by 16 publications

(18 citation statements)

References 13 publications

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“…. , N − 1, in (18), the master problem (18) is augmented with new constraints and is solved again. If (20) holds for all variables w N+j , and thus no new cuts are generated during an iteration, the current solution x * , w * of the master problem is optimal for the original LP approximation problem (16).…”

Section: A Cutting-plane Procedures For Polyhedral Approximations Of Pmentioning

confidence: 99%

“…reformulation (14), where the master problem has the form (18), and for a given solution x * , w * of the master, cuts of the form (24) are added if condition (25) is not satisfied for a specific j. Assuming that (18) is bounded, this cutting-plane procedure terminates after a finite number of iterations for any given ε > 0, with, perhaps, some anti-cycling scheme applied.…”

Section: Remark 4 An Example Of the Approximation Functionmentioning

confidence: 99%

“…Assuming that (18) is bounded, this cutting-plane procedure terminates after a finite number of iterations for any given ε > 0, with, perhaps, some anti-cycling scheme applied. In particular, the algorithm is guaranteed to generate at most O(ε −1 ) cutting planes, and in the special case of p = 2 the described cutting-plane algorithm is guaranteed to stop after at most O(ε −0.5 ) iterations.…”

Section: Remark 4 An Example Of the Approximation Functionmentioning

confidence: 99%

“…A polyhedral approximation approach to pOCP problems was considered by Krokhmal and Soberanis [15]. In the case when p is a rational number, the existing primal-dual methods of SOCP can be employed for solving p-order cone optimization problems using a reduction of p-order cone constraints to a system of linear and second-order cone constraints proposed by Nesterov and Nemirovski [21] and Ben-Tal and Nemirovski [8] (see also [18]). …”

Section: Introductionmentioning

confidence: 99%

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Polyhedral approximations inp-order cone programming

Vinel

Krokhmal

2014

Optimization Methods and Software

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This paper discusses the use of polyhedral approximations in solving p-order cone programming (pOCP) problems, or linear problems with p-order cone constraints, and their mixed-integer extensions. In particular, it is shown that the cutting-plane technique proposed in Krokhmal and Soberanis [Risk optimization with p-order conic constraints: A linear programming approach, Eur. J. Oper. Res. 201 (2010), pp. 653-671, http://dx.] for a special type of polyhedral approximations of pOCP problems, which allows for generation of cuts in constant time not dependent on the accuracy of approximation, is applicable to a larger family of polyhedral approximations. We also show that it can further be extended to form an exact solution method for pOCP problems with O(ε −1 ) iteration complexity. Moreover, it is demonstrated that an analogous constant-time cut-generating algorithm exists for recursively constructed lifted polyhedral approximations of second-order cones due to Ben-Tal and Nemirovski [On polyhedral approximations of the second-order cone, Math. Oper. Res. 26 (2001), pp. 193-205. Available at http://dx.doi.org/10.1287. It is also shown that the developed polyhedral approximations and the corresponding cutting-plane solution methods can be efficiently used for obtaining exact solutions of mixed-integer pOCP problems.

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Section: A Cutting-plane Procedures For Polyhedral Approximations Of Pmentioning

confidence: 99%

Section: Remark 4 An Example Of the Approximation Functionmentioning

confidence: 99%

Section: Remark 4 An Example Of the Approximation Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polyhedral approximations inp-order cone programming

Vinel

Krokhmal

2014

Optimization Methods and Software

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“…Such a representation, however, is not unique and, in general, may comprise a varying number of rotated second-order cones for a given p = q/s. In this case study, we use the technique of Morenko et al (2013), which allows for representing rational order p-cones with p = q/s in N +1 via N log 2 q second-order cones. Namely, in the case of p = 3, when q = 3, s = 1, the three-order cone (30) can equivalently be replaced with log 2 3 N = 2N quadratic cones…”

Section: Socp Reformulation Of P-ordermentioning

confidence: 99%

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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