Cone Programming Approach Research Articles

Robust portfolio optimization: a conic programming approach

The Markowitz Mean Variance model (MMV) and its variants are widely used for portfolio selection. The mean and covariance matrix used in the model originate from probability distributions that need to be determined empirically. It is well known that these parameters are notoriously difficult to estimate. In addition, the model is very sensitive to these parameter estimates. As a result, the performance and composition of MMV portfolios can vary significantly with the specification of the mean and covariance matrix. In order to address this issue we propose a one-period mean-variance model, where the mean and covariance matrix are only assumed to belong to an exogenously specified uncertainty set. The robust mean-variance portfolio selection problem is then written as a conic program that can be solved efficiently with standard solvers. Both second order cone program (SOCP) and semidefinite program (SDP) formulations are discussed. Using numerical experiments with real data we show that the portfolios generated by the proposed robust mean-variance model can be computed efficiently and are not as sensitive to input errors as the classical MMV’s portfolios.

Robust portfolio selection based on a joint ellipsoidal uncertainty set

‘Separable’ uncertainty sets have been widely used in robust portfolio selection models (e.g. see [E. Erdoğan, D. Goldfarb, and G. Iyengar, Robust portfolio management, manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004; D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38; R.H. Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]). For these uncertainty sets, each type of uncertain parameter (e.g. mean and covariance) has its own uncertainty set. As addressed in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-z], these ‘separable’ uncertainty sets typically share two common properties: (1) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set, is unknown, and it can be much higher than the desired one; and (2) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified, as observed in the computational experiments conducted in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z; R.H.Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]. To combat these drawbacks, we consider a factor model for random asset returns. For this model, we introduce a ‘joint’ ellipsoidal uncertainty set for the model parameters and show that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. We further show that the robust maximum risk-adjusted return (RMRAR) problem with this uncertainty set can be reformulated and solved as a cone programming problem. The computational results reported in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z] demonstrate that the robust portfolio determined by the RMRAR model with our ‘joint’ uncertainty set outperforms that with Goldfarb and Iyengar’s ‘separable’ uncertainty set proposed in the seminal paper [D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38] in terms of wealth growth rate and transaction cost; moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Support vector machine classification with noisy data: a second order cone programming approach

In this paper, we investigate the stability of linear and quadratic programming support vector machines (SVMs) with bounded noise in the input data using a robust optimisation model. For a linear discriminant function, this model is expressed as a second order cone optimisation problem. Using the concept of the kernel function, we generalise for nonlinear discriminant functions. Intuitively, it looks quite clear that large margin classifiers are robust in terms of bounded input noise. However, there is no theoretical analysis investigating this behaviour. We show that the SVM solution is stable under bounded perturbations of the data both in the linear programming and quadratic programming. Computational results are also presented for toy and real-world data.

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise-linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst-case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box-type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst-case risk under distributional ambiguity. New closed-form results for the worst-case optimized certainty equivalent risk measures and optimal portfolios are provided for two- and three-piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.

A Sequential Conic Programming Approach for the Coordinated and Robust Design of Power System Stabilizers

This paper shows that conic programming is an effective tool to solve robust power system stabilizer (PSS) design problems, namely coordinated gain tuning and coordinated phase and gain tuning. Design robustness is achieved by simultaneously considering several operating scenarios. The method is implemented through a sequence of conic programming runs that define a multivariable root locus along which the eigenvalues move. Specifically, the eigenvalues corresponding to the unstable and poorly damped modes are moved to a conic sector in the left half of the s-plane, whereas the eigenvalues corresponding to the well damped modes are constrained to stay within the boundaries of this conic sector. At each step of the solution, the PSS design parameters are restricted in a trust-region such that the computation of the eigenvalue shift based on the residue method holds valid. The proposed method is demonstrated on a 68-bus test system with nine different operating conditions. Comparisons are carried out between conic programming implementations for PSS coordinated gain tuning and for simultaneous tuning of gain and phase characteristics.

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

We propose a new method to compute the joint spectral radius and the joint spectral subradius of a set of matrices. We first restrict our attention to matrices that leave a cone invariant. The accuracy of our algorithm, depending on geometric properties of the invariant cone, is estimated. We then extend our method to arbitrary sets of matrices by a lifting procedure, and we demonstrate the efficiency of the new algorithm by applying it to several problems in combinatorics, number theory, and discrete mathematics.

Computing closest saddle node bifurcations in a radial system via conic programming

This paper considers the problem of computing the loading limits in a radial system which are (i) locally closest to current operating load powers and (ii) at which saddle node bifurcation occurs. The procedure is based on a known technique which requires iterating between two computational steps until convergence. In essence, step 1 produces a vector normal to the real and/or reactive load solution space boundary, whereas step 2 computes the bifurcation point along that vector. The paper shows that each of the above computational steps can be formulated as a second-order cone program for which polynomial time interior-point methods and efficient implementations exist. The proposed conic programming approach is used to compute the closest bifurcation points and the corresponding worst case load power margins of eleven different distribution systems. The approach is validated graphically and the existence of multiple load power margins is investigated.

A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set

The “separable” uncertainty sets have been widely used in robust portfolio selection models [e.g., see Erdogan et al. (Robust portfolio management. manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004), Goldfarb and Iyengar (Math Oper Res 28:1–38, 2003), Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004)]. For these uncertainty sets, each type of uncertain parameters (e.g., mean and covariance) has its own uncertainty set. As addressed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008), these “separable” uncertainty sets typically share two common properties: (i) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set is unknown, and it can be much higher than the desired one; and (ii) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified as observed in the computational experiments conducted in this paper and Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004). To combat these drawbacks, the author of this paper introduced a “joint” ellipsoidal uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and showed that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. For this uncertainty set, we showed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) that the corresponding robust maximum risk-adjusted return (RMRAR) model can be reformulated and solved as a cone programming problem. In this paper, we conduct computational experiments to compare the performance of the robust portfolios determined by the RMRAR models with our “joint” uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and Goldfarb and Iyengar’s “separable” uncertainty set proposed in the seminal paper (Goldfarb and Iyengar in Math Oper Res 28:1–38, 2003). Our computational results demonstrate that our robust portfolio outperforms Goldfarb and Iyengar’s in terms of wealth growth rate and transaction cost, and moreover, ours is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems

High peak-to-average power ratio (PAPR) is a critical issue in any multicarrier systems using orthogonal frequency division multiplexing, as terrestrial digital video broadcasting (DVB). It can result in low power efficiency and large performance degradation of systems, due to the nonlinearity of high-power amplifier (HPA). A PAPR reduction method based on tone reservation technique with second-order cone programming (SOCP) approach in terrestrial DVB systems is proposed. The authors first demonstrate the superiority of the SOCP optimisation algorithm compared with an iterative gradient-based algorithm, using the current DVB-T parameters: significant PAPR reduction gains can be achieved with only a very small set of subcarriers in the useful bandwidth, making the proposed method more promising in terms of spectral efficiency. Moreover, the proposed solution presents a very good trade-off between PAPR reduction gain and mean transmitted power increase. An overall study, taking into account the limitation of the power level of the dedicated subcarriers and the evaluation of the performances in presence of a nonlinear HPA, is presented. These performances are given in terms of adjacent channel power ratio and bit error rate. The resulting PAPR reduction gain demonstrates that the relevance of the proposed method for the future DVB-T standard is straightforward.

Counterweight Balancing for Vibration Reduction of Elastically Mounted Machine Frames: A Second-Order Cone Programming Approach

A moving linkage exerts fluctuating forces and moments on its supporting frame. One strategy to suppress the resulting frame vibration is to reduce the exciting forces and moments by adding counterweights to the linkage links. This paper develops a generic methodology to design such counterweights for planar linkages, based on formulating counterweight design as a second-order cone program. Second-order cone programs are convex, which implies that these nonlinear optimization problems have a global optimum that is guaranteed to be found in a numerically efficient manner. Two optimization criteria are considered: the frame vibration itself and the dynamic force transmitted to the machine floor. While the methodology is valid regardless of the complexity of the considered linkage, it is developed here for a literature benchmark consisting of a crank-rocker four-bar linkage supported by a rigid, elastically mounted frame with three degrees of freedom. For this particular benchmark, the second-order cone program slightly improves the previously known optimum. Moreover, numerical comparison with current state-of-the-art algorithms for nonlinear optimization shows that our approach results in a substantial reduction of the required computational time.

Support vector regression with noisy data: a second order cone programming approach

A robust convex optimization approach is proposed for support vector regression (SVR) with noisy input data. The data points are assumed to be uncertain, but bounded within given hyper-spheres of radius η. The proposed Robust SVR model is equivalent to a Second Order Cone Programming (SOCP) problem. SOCP formulation with Gaussian noise models assumption is discussed. Computational results are presented both on real world and synthetic data sets. The robust SOCP approach is compared with several other regression algorithms such as SVR, least-square SVR, and artificial neural networks by injecting Gaussian noise to each of the data points. The proposed approach out performs the other regression algorithms for some data sets. Moreover, the generalization behavior of the SOCP method is better than the traditional SVR with increasing the uncertainty level η until a threshold value.

Conic programming approach for static voltage stability analysis in radial networks

That the problem of computing the capacity limit of a radial distribution system can be formulated as a second-order cone program is shown. The implications of the conic programming formulation are 2-fold. First, the load capability of the radial system can be obtained using existing efficient implementations of polynomial time interior-point algorithms, thus avoiding the need for running a sequence of load flow solutions. Secondly, the conic objective function yields a voltage stability indicator (SI). This indicator quantifies the maximum percentage by which the current load profile can be uniformly increased before voltage collapse occurs. The proposed method is validated by computing the load capability and voltage SIs of 11 different distribution systems. Comparisons are carried out with five previously published voltage SIs.

MAP-Based Underdetermined Blind Source Separation of Convolutive Mixtures by Hierarchical Clustering and -Norm Minimization

We address the problem of underdetermined BSS. While most previous approaches are designed for instantaneous mixtures, we propose a time-frequency-domain algorithm for convolutive mixtures. We adopt a two-step method based on a general maximum a posteriori (MAP) approach. In the first step, we estimate the mixing matrix based on hierarchical clustering, assuming that the source signals are suciently sparse. The algorithm works directly on the complex-valued data in the time-frequency domain and shows better convergence than algorithms based on self-organizing maps. The assumption of Laplacian priors for the source signals in the second step leads to an algorithm for estimating the source signals. It involves the l1-norm minimization of complex numbers because of the use of the time-frequency-domain approach. We compare a combinatorial approach initially designed for real numbers with a second-order cone programming (SOCP) approach designed for complex numbers. We found that although the former approach is not theoretically justified for complex numbers, its results are comparable to, or even better than, the SOCP solution. The advantage is a lower computational cost for problems with low input/output dimensions.

Robust linear receivers for multiaccess space-time block-coded MIMO systems: a probabilistically constrained approach

Traditional multiuser receiver algorithms developed for multiple-input-multiple-output (MIMO) wireless systems are based on the assumption that the channel state information (CSI) is precisely known at the receiver. However, in practical situations, the exact CSI may be unavailable because of channel estimation errors and/or outdated training. In this paper, we address the problem of robustness of multiuser MIMO receivers against imperfect CSI and propose a new linear technique that guarantees the robustness against CSI errors with a certain selected probability. The proposed receivers are formulated as probabilistically constrained stochastic optimization problems. Provided that the CSI mismatch is Gaussian, each of these problems is shown to be convex and to have a unique solution. The fact that the CSI mismatch is Gaussian also enables to convert the original stochastic problems to a more tractable deterministic form and to solve them using the second-order cone programming approach. Numerical simulations illustrate an improved robustness of the proposed receivers against CSI errors and validate their better flexibility as compared with the robust multiuser MIMO receivers based on the worst case designs.

Optimal design and verification of temporal and spatial filters using second-order cone programming approach

Temporal filters and spatial filters are widely used in many areas of signal processing. A number of optimal design criteria to these problems are available in the literature. Various computational techniques are also presented to optimize these criteria chosen. There are many drawbacks in these methods. In this paper, we introduce a unified framework for optimal design of temporal and spatial filters. Most of the optimal design problems of FIR filters and beamformers are included in the framework. It is shown that all the design problems can be reformulated as convex optimization form as the second-order cone programming (SOCP) and solved efficiently via the well-established interior point methods. The main advantage of our SOCP approach as compared with earlier approaches is that it can include most of the existing methods as its special cases, which leads to more flexible designs. Furthermore, the SOCP approach can optimize multiple required performance measures, which is the drawback of earlier approaches. The SOCP approach is also developed to optimally design temporal and spatial two-dimensional filter and spatial matrix filter. Numerical results demonstrate the effectiveness of the proposed approach.

A Conic Programming Approach to Generalized Tchebycheff Inequalities

Consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments. This is a special class of generalized Tchebycheff inequalities in probability theory. We study this problem within the framework of conic programming. Relying on a general approximation scheme for conic programming, we show that these bounds can be numerically computed or approximated via semidefinite programming. We also describe some applications of this class of generalized Tchebycheff inequalities in probability, finance, and inventory theory.

Design of practically perfect-reconstruction cosine-modulated filter banks: a second-order cone programming approach

Designing optimal perfect-reconstruction (PR) and near PR (NPR) cosine-modulated filter banks is essentially a constrained nonlinear minimization problem. This paper proposes two second-order cone-programming based algorithms for designing NPR and practically PR cosine-modulated filter banks with improved performance relative to several established design methods.

Adaptive beamforming with sidelobe control: a second-order cone programming approach

A new approach to adaptive beamforming with sidelobe control is developed. The proposed beamformer represents a modification of the popular minimum variance distortionless response (MVDR) beamformer. It minimizes the array output power while maintaining the distortionless response in the direction of the desired signal and a sidelobe level that is strictly guaranteed to be lower than some given (prescribed) threshold value. The resulting modified MVDR problem is shown to be convex, and its second-order cone (SOC) formulation is obtained that facilitates a computationally efficient way to implement our beamformer using the interior point method.

Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach

Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.