Cone Programming Problem Research Articles

Peak reduction in OFDM using second-order cone programming relaxation

In this paper, we address the problem of peak-to-average power ratio (PAPR) reduction in orthogonal frequency-division multiplexing (OFDM) systems. We formulate the problem as an error vector magnitude (EVM) optimization task with constraints on PAPR and free carrier power overhead (FCPO). This problem, which is known to be NP hard, is shown to be approximated by a second-order cone programming (SOCP) problem using a sequential convex programming approach, making it much easier to handle. This approach can be extended to the more general problem when PAPR, EVM, and FCPO are constrained simultaneously. Our performance results show the effectiveness of the proposed approach, which allows good performance with lower computational complexity and infeasibility rate than state-of-the-art PAPR-reduction convex approaches. Moreover, in the case when all the three system parameters are constrained simultaneously, the proposed approach outperforms the convex approaches in terms of infeasibility rate, PAPR, and bit error rate (BER) performances.

Learning Imbalanced Classifiers Locally and Globally with One-Side Probability Machine

We consider the imbalanced learning problem, where the data associated with one class are far fewer than those associated with the other class. Current imbalanced learning methods often handle this problem by adapting certain intermediate parameters so as to impose a bias on the minority data. However, most of these methods are in rigorous and need to adapt those factors via the trial-and-error procedure. Recently, a new model called Biased Minimax Probability Machine (BMPM) presents a rigorous and systematic work and has demonstrated very promising performance on imbalance learning. Despite its success, BMPM exclusively relies on global information, namely, the first order and second order data information; such information might be however unreliable, especially for the minority data. In this paper, we propose a new model called One-Side Probability Machine (OSPM). Different from the previous approaches, OSPM can lead to rigorous treatment on biased classification tasks. Importantly, the proposed OSPM exploits the reliable global information from one side only, i.e., the majority class, while engaging the robust local learning from the other side, i.e., the minority class. To our best knowledge, OSPM presents the first model capable of learning data both locally and globally. Our proposed model has also established close connections with various famous models such as BMPM, Support Vector Machine, and Maxi-Min Margin Machine. One appealing feature is that the optimization problem involved in the novel OSPM model can be cast as a convex second order conic programming problem with the global optimum guaranteed. A series of experimental results on three data sets demonstrate the advantages of our proposed methods over four competitive approaches.

A note on the minimal cone for conic linear programming

In a recent paper by the author (4OR, 9:403–416, 2011), a recursive procedure to obtain the minimal cone for the constraint system of a conic linear programming problem has been developed. In this short note, we point out some minor errors for this recursive procedure, make corrections of these errors, and present some interesting results pertaining to the recursive process.

Conic approximation to nonconvex quadratic programming with convex quadratic constraints

In this paper, a conic reformulation and approximation is proposed for solving a nonconvex quadratic programming problem subject to several convex quadratic constraints. The original problem is transformed into a linear conic programming problem, which can be approximated by a sequence of linear conic programming problems over the dual cone of the cone of nonnegative quadratic functions. Since the dual cone of the cone of nonnegative quadratic functions has a linear matrix inequality representation, each linear conic programming problem in the sequence can be solved efficiently using the semidefinite programming techniques. In order to speed up the convergence of the approximation sequence and relieve the computational effort in solving the linear conic programming problems, an adaptive scheme is adopted in the proposed algorithm. We prove that the lower bounds generated by the linear conic programming problems converge to the optimal value of the original problem. Several numerical examples are used to illustrate how the algorithm works and the computational results demonstrate the efficiency of the proposed algorithm.

Adaptive computable approximation to cones of nonnegative quadratic functions

Cones of nonnegative quadratic functions are keys to the understanding of quadratic optimization problems, since any quadratically constrained quadratic programming problem can be reformulated as a linear conic programming problem over such a cone. This paper proposes an adaptive computable approximation scheme to cones of nonnegative quadratic functions and uses it for solving linear conic programming problems over such a cone. We study some basic properties of cones of nonnegative quadratic functions and present a class of simple cones with computable linear matrix inequalities representations. Building on these simple cones, we design a computable approximation scheme for handling a general cone of nonnegative quadratic functions. When the scheme is applied for solving linear conic programming problems over a cone of nonnegative quadratic functions, we incorporate an adaptive approach to enhance the performance of the proposed computable approximation scheme. The computational performance and theoretic convergence proof of the proposed adaptive computable approximation scheme are shown for solving box-constrained quadratic programming problems.

Polyhedral approximations in p-order cone programming

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.doi.org/10.1016/j.ejor.2009.03.053] 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/moor.26.2.193.10561]. 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.

Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids

The paper is devoted to a numerical Limit Analysis of a hollow spheroidal model with a Drucker–Prager solid matrix, for several values of the corresponding friction angle ϕ. In the first part of this study, the static and the mixed kinematic 3D-codes recently evaluated in [1] are modified to use the geometry defined in [2] for spheroidal cavities in the context of a von Mises matrix. The results in terms of macroscopic criteria are satisfactory for low and medium values of ϕ, but not enough for ϕ=30° in the highly compressive part of the criterion. To improve these results, an original mixed approach, dedicated to the axisymmetric case, was elaborated with a specific discontinuous quadratic velocity field associated with the triangular finite element. Despite the less good conditioning inherent to the axisymmetric modelization, the resulting conic programming problem appears quite efficient, allowing one take into account numerical discretization refinements unreachable with the corresponding 3D mixed code. After a first validation in the case of spherical cavities whose exact solution is known, the final results for spheroidal voids are given for three usual values of the friction angle and two values of the cavity aspect factor.

Regularity conditions characterizing Fenchel–Lagrange duality and Farkas-type results in DC infinite programming

In this paper, we consider a DC infinite programming problem (P) with inequality constraints. By using the properties of the epigraph of the conjugate functions, we introduce some new notions of regularity conditions for (P). Under these new regularity conditions, we completely characterize the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for (P). Similarly, we also completely characterize the Farkas-type results and the stable Farkas-type results for (P). As applications, we obtain the corresponding results for conic programming problems.

Convex Optimization inR

Convex optimization now plays an essential role in many facets of statistics. We briefly survey some recent developments and describe some implementations of these methods in R . Applications of linear and quadratic programming are introduced including quantile regression, the Huber M-estimator and various penalized regression methods. Applications to additively separable convex problems subject to linear equality and inequality constraints such as nonparametric density estimation and maximum likelihood estimation of general nonparametric mixture models are described, as are several cone programming problems. We focus throughout primarily on implementations in the R environment that rely on solution methods linked to R, like MOSEK by the package Rmosek. Code is provided in R to illustrate several of these problems. Other applications are available in the R package REBayes, dealing with empirical Bayes estimation of nonparametric mixture models.

A branch-and-cut approach to portfolio selection with marginal risk control in a linear conic programming framework

Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio. In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. The optimization model involved is a nonconvex quadratically constrained quadratic programming (QCQP) problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming (SDP) relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an ∈-optimal global solution (with respect to feasibility and optimality) in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.

Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems

In this paper, we present an optimality condition which could determine whether a given KKT solution is globally optimal. This condition is equivalent to determining if the Hessian of the corresponding Largrangian is copositive over a set. To find the corresponding Lagrangian multiplier, two linear conic programming problems are constructed and then relaxed for computational purpose. Under the new condition, we proposed a local search based scheme to find a global optimal solution and showed its effectiveness by three examples.

Quadratic optimization over one first-order cone

This paper studies the first-order cone constrained homogeneous quadratic programming problem. For efficient computation, the problem is reformulated as a linear conic programming problem. A union of second-order cones are designed to cover the first-order cone such that a sequence of linear conic programming problems can be constructed to approximate the conic reformulation. Since the cone of nonnegative quadratic forms over a union of second-order cones has a linear matrix inequalities representation, each linear conic programming problem in the sequence is polynomial-time solvable by applying semidefinite programming techniques. The convergence of the sequence is guaranteed when the union of second-order cones gets close enough to the first-order cone. In order to further improve the efficiency, an adaptive scheme is adopted. Numerical experiments are provided to illustrate the efficiency of the proposed approach.

DUALITY AND FARKAS-TYPE RESULTS FOR DC INFINITE PROGRAMMING WITH INEQUALITY CONSTRAINTS

In this paper, a DC infinite programming problem with inequality constraints is considered. By using the method of Fenchel conjugate functions, a dual scheme for the DC infinite programming problem is introduced. Then, under suitable conditions, weak and strong duality assertions are obtained. Moreover, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of the DC infinite programming problem are given. As applications, the proposed approach is applied to conic programming problems.

Damper placement optimization in a shear building model with discrete design variables: a mixed‐integer second‐order cone programming approach

SUMMARYSupplemental damping is known as an efficient and practical means to improve seismic response of building structures. Presented in this paper is a mixed‐integer programming approach to find the optimal placement of supplemental dampers in a given shear building model. The damping coefficients of dampers are treated as discrete design variables. It is shown that a minimization problem of the sum of the transfer function amplitudes of the interstory drifts can be formulated as a mixed‐integer second‐order cone programming problem. The global optimal solution of the optimization problem is then found by using a solver based on a branch‐and‐cut algorithm. Two numerical examples in literature are solved with discrete design variables. In one of these examples, the proposed method finds a better solution than an existing method in literature developed for the continuous optimal damper placement problem. Copyright © 2013 John Wiley & Sons, Ltd.

Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme

It is co-NP-complete to decide whether a given matrix is copositive or not. In this paper, this decision problem is transformed into a quadratic programming problem, which can be approximated by solving a sequence of linear conic programming problems defined on the dual cone of the cone of nonnegative quadratic functions over the union of a collection of ellipsoids. Using linear matrix inequalities (LMI) representations, each corresponding problem in the sequence can be solved via semidefinite programming. In order to speed up the convergence of the approximation sequence and to relieve the computational effort of solving linear conic programming problems, an adaptive approximation scheme is adopted to refine the union of ellipsoids. The lower and upper bounds of the transformed quadratic programming problem are used to determine the copositivity of the given matrix.

The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras

Lowner’s operator in Euclidean Jordan algebras, defined via the spectral decomposition of the elements of a scalar function, has been widely used in various optimization problems over Euclidean Jordan algebras. In this note, we shall show that Lowner’s operator in Euclidean Jordan algebras is Holder continuous if and only if the underlying scalar function is Holder continuous. Such a property will be useful in designing solution methods for symmetric cone programming and symmetric cone complementarity problem.

A Strong Dual for Conic Mixed-Integer Programs

Mixed-integer conic programming is a generalization of mixed-integer linear programming. In this paper, we present an extension of the duality theory for mixed-integer linear programming (see [M. Guzelsoy and T. K. Ralphs, Int. J. Oper. Res. (Taichung), 4 (2007), pp. 118--137], [G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience, New York, 1988]) to the case of mixed-integer conic programming. In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we show that strong duality holds.

Square-root lasso: pivotal recovery of sparse signals via conic programming

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors $p$ is large, possibly much larger than $n$, but only $s$ regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation $\sigma$ or nor does it need to pre-estimate $\sigma$. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate $\sigma \{(s/n)\log p\}^{1/2}$ in the prediction norm, and thus matching the performance of the lasso with known $\sigma$. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.

KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems

To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.

Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ or nor does it need to pre-estimate σ. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate σ{(s/n) log p}1/2 in the prediction norm, and thus matching the performance of the lasso with known σ. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.