Conic Optimization Problems Research Articles

Numerical limit analysis of reinforced concrete slabs using a dual approach and conic programming

This study presents a numerical procedure for the analysis of reinforced concrete slabs (RCS) that obey Nielsen's yield criterion. A pseudo lower bound formulation of finite elements is established to solve the static theorem as a conic optimization problem with the aim to determine the maximum load capacity of RCS. Lower bound solution is improved by means of an adaptive remeshing strategy using a dissipation estimator which is obtained on the normalization of the kinematic criterion defined from the yield criteria. Different known examples are evaluated and the results show a good agreement with the exact values of the collapse load as well as numerical convergence. The displacement rates are estimated from the dual solution established in the optimization problem in order to determine the collapse mechanism in the slab.

A relaxed-certificate facial reduction algorithm based on subspace intersection

A “facial reduction”-like regularization algorithm is established for general conic optimization problems by relaxing requirements on the reduction certificates. This yields a rapid subspace reduction algorithm challenged only by representational issues of the regularized cone. A condition for practical usage is analyzed and shown to always be satisfied for single second-order cone optimization problems. Should the condition fail on some other class of instances, only partial regularization is achieved based on the success of the individual subspace intersection.

Efficient formulations of the material identification problem using full-field measurements

The material identification problem addressed consists of determining the constitutive parameters distribution of a linear elastic solid using displacement measurements. This problem has been considered in important applications such as the design of methodologies for breast cancer diagnosis. Since the resolution of real life problems involves high computational costs, there is great interest in the development of efficient methods. In this paper two new efficient formulations of the problem are presented. The first formulation leads to a second-order cone optimization problem, and the second one leads to a quadratic optimization problem, both allowing the resolution of the problem with high efficiency and precision. Numerical examples are solved using synthetic input data with error. A regularization technique is applied using the Morozov criterion along with an automatic selection strategy of the regularization parameter. The proposed formulations present great advantages in terms of efficiency, when compared to other formulations that require the application of general nonlinear optimization algorithms.

An efficient inexact symmetric Gauss–Seidel based majorized ADMM for high-dimensional convex composite conic programming

In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an inexact sGS method, instead of an unknown number of cycles, to solve each of the subproblems involved.With some simple and implementable error tolerance criteria, the cost for solving the subproblems can be greatly reduced, and many steps in the forward sweep of each sGS cycle can often be skipped, which further contributes to the efficiency of the proposed method. Global convergence as well as the iteration complexity in the non-ergodic sense is established.Preliminary numerical experiments on some high-dimensional linear and convex quadratic SDP problems with a large number of linear equality and inequality constraints are also provided. The results show that for the vast majority of the tested problems, the sGS-imsPADMM is 2 to 3 times faster than the directly extended multi-block ADMM with the aggressive step-length of 1.618, which is currently the benchmark among first-order methods for solving multi-block linear and quadratic SDP problems though its convergence is not guaranteed.

A Second Order Cone Formulation of Continuous CTA Model.

In this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using ℓ1 or ℓ2 norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the ℓ1-CTA using Pseudo-Huber function was introduced in an attempt to combine positive characteristics of both ℓ1-CTA and ℓ2-CTA. All three models can be solved using appropriate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic optimization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and ℓ1-CTA as Second-Order Cone (SOC) optimization problems and test the validity of the approach on the small example of two-dimensional tabular data set.

Local Superlinear Convergence of Polynomial-Time Interior-Point Methods for Hyperbolicity Cone Optimization Problems

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have negative curvature. We propose a new path-following predictor-corrector scheme, which works only in the dual space. It is based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local superlinear one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size maintaining feasibility. As the optimal solution set is approached, our algorithm automatically tightens the neighborhood of the central path proportionally to the current duality gap.

Outage Based Robust Relay Beamforming Problem in Multi-User Networks

In this paper, a robust relay beamforming problem has been dealt with by using the stochastic approach with imperfect knowledge of channel state information. Specifically, we have modeled the channel estimation error as a Gaussian random variable. Our objective is to minimize the total relay's transmit power subject to outage SINR constraint criterion at each receiver. We aim to establish approximate probabilistic SINR constrained formulations in the form of convex conic optimization problem. It is shown that the original robust problem is a nonconvex problem, but it can be relaxed to an SDP convex problem using the semi-definite relaxation technique. In this paper, we have compared our stochastic method with the worst-case robust method in terms of power consumption efficiency and complexity. Simulation results verify that our robust method outperforms the worst case approach.

A polynomial-time interior-point method for circular cone programming based on kernel functions

We present an interior-point method based on kernel functions for circular cone optimization problems, which has been found useful for describing optimal design problems ofoptimal grasping manipulation for multi-fingered robots. Since the well-known second order cone is a particular circular cone, we first establish an invertible linear mapping between a circular cone and its corresponding second order cone. Then we develop akernel function based interior-point method to solve circular cone optimization in terms of the corresponding second order cone optimization problem.We derive the complexity bound of the interior-point method and conclude that circular cone optimization ispolynomial-time solvable. Finally we illustrate the performance of interior-point method bya real-world quadruped robot example of optimal contact forces taken from the literature [10].

Application of a GPU-accelerated hybrid preconditioned conjugate gradient approach for large 3D problems in computational geomechanics

This paper presents a hybrid preconditioning technique for Conjugate Gradient method and discusses its parallel implementation on Graphic Processing Unit (GPU) for solving large sparse linear systems arising from application of interior point methods to conic optimization problems in the context of nonlinear Finite Element Limit Analysis (FELA) for computational Geomechanics. For large 3D problems, the use of direct solvers in general becomes prohibitively expensive due to exponentially growing memory requirements and computational time. Besides, the so-called saddle-point systems resulting from use of optimization framework is not an exemption. On the other hand, although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods, they usually exhibit high number of iterations to reach convergence. In present paper, we show that this problem can be effectively tackled using the proposed hybrid preconditioner along with an elaborate implementation on GPU. Furthermore, numerical results verify the robustness and efficiency of the proposed technique.

Primal-dual interior-point algorithms for convex quadratic circular cone optimization

In this paper we focus on a class of special nonsymmetric cone optimization problem called circular cone optimization problem, which has a convex quadratic function as the objective function and an intersection of a non-self-dual circular cone and linear equations as the constraint condition. Firstly we establish the algebraic relationships between the circular cone and the second-order cone and translate the original problem from the circular cone optimization problem to the second-order cone optimization problem.Then we present kernel-function based primal-dual interior-point algorithms for solving this special circular cone optimizationand derive the iteration bounds for large- and small-update methods.Finally, some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithms.

Outage Constrained Robust Transmit Optimization for Multiuser MISO Downlinks: Tractable Approximations by Conic Optimization

In this paper we consider a probabilistic signal-to-interference and-noise ratio (SINR) constrained problem for transmit beamforming design in the presence of imperfect channel state information (CSI), under a multiuser multiple-input single-output (MISO) downlink scenario. In particular, we deal with outage-based quality-of-service constraints, where the probability of each user's SINR not satisfying a service requirement must not fall below a given outage probability specification. The study of solution approaches to the probabilistic SINR constrained problem is important because CSI errors are often present in practical systems and they may cause substantial SINR outages if not handled properly. However, a major technical challenge is how to process the probabilistic SINR constraints. To tackle this, we propose a novel relaxation- restriction (RAR) approach, which consists of two key ingredients-semidefinite relaxation (SDR), and analytic tools for conservatively approximating probabilistic constraints. The underlying goal is to establish approximate probabilistic SINR constrained formulations in the form of convex conic optimization problems, so that they can be readily implemented by available solvers. Using either an intuitive worst-case argument or specialized probabilistic results, we develop various conservative approximation schemes for processing probabilistic constraints with quadratic uncertainties. Consequently, we obtain several RAR alternatives for handling the probabilistic SINR constrained problem. Our techniques apply to both complex Gaussian CSI errors and i.i.d. bounded CSI errors with unknown distribution. Moreover, results obtained from our extensive simulations show that the proposed RAR methods significantly improve upon existing ones, both in terms of solution quality and computational complexity.

Kernel-function-based primal-dual interior-point methods for convex quadratic optimization over symmetric cone

Abstract In this paper, we give a unified computational scheme for the complexity analysis of kernel-function-based primal-dual interior-point methods for convex quadratic optimization over symmetric cone. By using Euclidean Jordan algebras, the currently best-known iteration bounds for large- and small-update methods are derived, namely, O ( r log r log r ε ) and O ( r log r ε ) , respectively. Furthermore, this unifies the analysis for a wide class of conic optimization problems. MSC:90C25, 90C51.

Robust solutions to multi-facility Weber location problem under interval and ellipsoidal uncertainty

In this paper, we consider the multi-facility Weber location problem (MFWP) with uncertain location of demand points and transportation cost parameters. Equivalent formulations of its robust counterparts for both the Euclidean and block norms and interval and ellipsoidal uncertainty sets are given as conic linear optimization problems.

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

A homogeneous interior-point algorithm for solving nonsymmetric convex conic optimization problems is presented. Starting each iteration from the vicinity of the central path, the method steps in the approximate tangent direction and then applies a correction phase to locate the next well-centered primal–dual point. Features of the algorithm include that it makes use only of the primal barrier function, that it is able to detect infeasibilities in the problem and that no phase-I method is needed. We prove convergence to $$\epsilon $$ -accuracy in $${\mathcal {O}}(\sqrt{\nu } \log {(1/\epsilon )})$$ iterations. To improve performance, the algorithm employs a new Runge–Kutta type second order search direction suitable for the general nonsymmetric conic problem. Moreover, quasi-Newton updating is used to reduce the number of factorizations needed, implemented so that data sparsity can still be exploited. Extensive and promising computational results are presented for the $$p$$ -cone problem, the facility location problem, entropy maximization problems and geometric programs; all formulated as nonsymmetric convex conic optimization problems.

Robust two-stage stochastic linear programs with moment constraints

We consider the two-stage stochastic linear programming model, in which the recourse function is a worst case expected value over a set of probabilistic distributions. These distributions share the same first- and second-order moments. By using duality of semi-infinite programming and assuming knowledge on extreme points of the dual polyhedron of the constraints, we show that a deterministic equivalence of the two-stage problem is a second-order cone optimization problem. Numerical examples are presented to show non-conservativeness and computational advantage of this approach.

Decomposition in Conic Optimization with Partially Separable Structure

Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators.

A Safe Approximation Approach to Secrecy Outage Design for MIMO Wiretap Channels

Consider a multi-input multi-output (MIMO) channel wiretapped by multiple multi-antenna eavesdroppers. Assuming imperfect eavesdroppers' channel state information (CSI) at the transmitter, an outage-constrained secrecy rate maximization (OC-SRM) problem is considered. Specifically, we aim to design the transmit covariance matrix such that the outage secrecy rate is maximized for a given outage probability. The OC-SRM problem is challenging, and as a compromise, we resort to a recently developed Bernstein-type inequality approach to obtain a safe (conservative) approximate solution for OC-SRM. The merit of the proposed safe design lies in its tractability. In particular, a safe solution can be efficiently computed by alternately solving two convex conic optimization problems. The efficacy of the proposed design is demonstrated by simulations.

Primal-Dual Subgradient Method for Huge-Scale Linear Conic Problems

In this paper, we develop a primal-dual subgradient method for solving huge-scale linear conic optimization problems. Our main assumption is that the primal cone is formed as a direct product of many small-dimensional convex cones and that the matrix $A$ of the corresponding linear operator is uniformly sparse. In this case, our method can approximate the primal-dual optimal solution with accuracy $\epsilon$ in $O\big({1 \over \epsilon^2}\big)$ iterations. At the same time, the complexity of each iteration of this scheme does not exceed $O(r q \log_2 n)$ operations, where $r$ and $q$ are the maximal numbers of nonzero elements in the rows and columns of matrix $A$, and $n$ is the number of variables.

On the Identification of the Optimal Partition of Second Order Cone Optimization Problems

This paper discusses the identification of the optimal partition of second order cone optimization (SOCO). By giving some condition numbers which only depend on the SOCO problem itself, we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition $\mathcal{B}, \mathcal{N}, \mathcal{R}$, and $\mathcal{T}$ for SOCO problems can be identified along the central path when the barrier parameter $\mu$ is small enough. Then we generalize the results to a specific neighborhood of the central path.

Handling Optimization under Uncertainty Problem Using Robust Counterpart Methodology

In this paper we discuss the robust counterpart (RC) methodology to handle the optimization under uncertainty problem as proposed by Ben-Tal and Nemirovskii. This optimization methodology incorporates the uncertain data in U a so-called uncertainty set and replaces the uncertain problem by its so-called robust counterpart. We apply the RC approach to uncertain Conic Optimization (CO) problems, with special attention to robust linear optimization (RLO) problem and include a discussion on parametric uncertainty for that case. Some new supported examples are presented to give a clear description of the used of RC methodology theorem.