Robust Semidefinite Programming Research Articles

Strong duality in parametric robust semi-definite linear programming and exact relaxations

This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.

Pareto robust optimization on Euclidean vector spaces

Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis in [Manage Sci 60(1):130–147, 9]. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with affine uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing a Pareto robustly optimal solution for a robust SDP is tractable and illustrate the benefit of such solutions at the example of the maximal eigenvalue problem. Furthermore, we modify the famous algorithm of Goemans and Williamson [Assoc Comput Mach 42(6):1115–1145, 8] in order to compute cuts for the robust max-cut problem that yield an improved approximation guarantee in non-worst-case scenarios.

Two-dimensional grid-less angle estimation based on three parallel nested arrays

An array geometry with three parallel nested arrays is designed and a corresponding two-dimensional (2D) grid-less angle estimation method is proposed. The array can use 4 M physical elements to achieve 4M2 virtual degrees of freedom (DOF) after the extended vectorization of two cross-covariance matrices (CCMs), which also transforms the 2D angle estimation problem into one-dimensional (1D) and one-snapshot sparse recovery. Thereafter, without any spatial smoothing, the Toeplitz covariance matrix of the virtual array can be recovered through a robust semi-definite programming (SDP) based on atomic norm minimization (ANM). Meanwhile, the denoising virtual output is obtained as a by-product. Finally, 1D angle is obtained from the Vandermonde decomposition of the Toeplitz matrix and the other 1D angle is obtained in succession from the denoising virtual output via total least squares (TLS). The proposed method is robust to grid mismatch problem, obtains automatically paired two-dimensional angle estimation and requires only one-level Toeplitz matrix recovery, which makes the maximum identifiable source number exceed the physical sensor number. Furthermore, the angle estimation performance of the proposed method outperforms other state-of-the-art methods using parallel arrays. Numerical simulations verify the effectiveness of our approach.

Interval Enclosures of Upper Bounds of Roundoff Errors Using Semidefinite Programming

A long-standing problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors. We present a framework to compute lower bounds on largest absolute roundoff errors, for a particular rounding model. This method applies to numerical programs implementing polynomial functions with box constrained input variables. Our study is based on three different hierarchies, relying respectively on generalized eigenvalue problems, elementary computations, and semidefinite programming (SDP) relaxations. This is complementary of over-approximation frameworks, consisting of obtaining upper bounds on the largest absolute roundoff error. Combining the results of both frameworks allows one to get enclosures for upper bounds on roundoff errors. The under-approximation framework provided by the third hierarchy is based on a new sequence of convergent robust SDP approximations for certain classes of polynomial optimization problems. Each problem in this hierarchy can be solved exactly via SDP. By using this hierarchy, one can provide a monotone nondecreasing sequence of lower bounds converging to the absolute roundoff error of a program implementing a polynomial function, applying for a particular rounding model. We investigate the efficiency and precision of our method on nontrivial polynomial programs coming from space control, optimization, and computational biology.

Robust Differential Received Signal Strength-Based Localization

Source localization based on signal strength measurements has become very popular due to its practical simplicity. However, the severe nonlinearity and non-convexity make the related optimization problem mathematically difficult to solve, especially when the transmit power or the path-loss exponent (PLE) is unknown. Moreover, even if the PLE is known but not perfectly estimated or the anchor location information is not accurate, the constructed data model will become uncertain, making the problem again hard to solve. This paper particularly focuses on differential received signal strength (DRSS)-based localization with model uncertainties in case of unknown transmit power and PLE. A new whitened model for DRSS-based localization with unknown transmit powers is first presented and investigated. When assuming the PLE is known, we introduce two estimators based on an exact data model, an advanced best linear unbiased estimator (A-BLUE) and a Lagrangian estimator (LE), and then we present a robust semidefinite programming (SDP)-based estimator (RSDPE), which can cope with model uncertainties (imperfect PLE and inaccurate anchor location information). The three proposed estimators have their own advantages from different perspectives: the A-BLUE has the lowest complexity; the LE holds the best accuracy for a small measurement noise; and the RSDPE yields the best performance under a large measurement noise and possesses a very good robustness against model uncertainties. Finally, we propose a robust SDP-based block coordinate descent estimator (RSDP-BCDE) to deal with a completely unknown PLE and its performance converges to that of the RSDPE using a perfectly known PLE.

Applications of gauge duality in robust principal component analysis and semidefinite programming

Gauge duality theory was originated by Freund [Math. Programming, 38(1):47-67, 1987] and was recently further investigated by Friedlander, Mac{\^e}do and Pong [SIAM J. Optm., 24(4):1999-2022, 2014]. When solving some matrix optimization problems via gauge dual, one is usually able to avoid full matrix decompositions such as singular value and/or eigenvalue decompositions. In such an approach, a gauge dual problem is solved in the first stage, and then an optimal solution to the primal problem can be recovered from the dual optimal solution obtained in the first stage. Recently, this theory has been applied to a class of \emph{semidefinite programming} (SDP) problems with promising numerical results [Friedlander and Mac{\^e}do, SIAM J. Sci. Comp., to appear, 2016]. In this paper, we establish some theoretical results on applying the gauge duality theory to robust \emph{principal component analysis} (PCA) and general SDP. For each problem, we present its gauge dual problem, characterize the optimality conditions for the primal-dual gauge pair, and validate a way to recover a primal optimal solution from a dual one. These results are extensions of [Friedlander and Mac{\^e}do, SIAM J. Sci. Comp., to appear, 2016] from nuclear norm regularization to robust PCA and from a special class of SDP which requires the coefficient matrix in the linear objective to be positive definite to SDP problems without this restriction. Our results provide further understanding in the potential advantages and disadvantages of the gauge duality theory.

Robust least square semidefinite programming with applications

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196---1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.

Strong duality in robust semi-definite linear programming under data uncertainty

This article develops the deterministic approach to duality for semi-definite linear programming problems in the face of data uncertainty. We establish strong duality between the robust counterpart of an uncertain semi-definite linear programming model problem and the optimistic counterpart of its uncertain dual. We prove that strong duality between the deterministic counterparts holds under a characteristic cone condition. We also show that the characteristic cone condition is also necessary for the validity of strong duality for every linear objective function of the original model problem. In addition, we derive that a robust Slater condition alone ensures strong duality for uncertain semi-definite linear programs under spectral norm uncertainty and show, in this case, that the optimistic counterpart is also computationally tractable.

Robust linear transceivers for downlink multi-user multiple-input multiple-output systems using second-order cone programming optimisation

This study addresses the joint robust linear transceiver design problems for a downlink multi-user multiple-input multiple-output (MIMO) antenna system in the presence of imperfect channel state information (CSI). The uncertainty in the channel is characteried by a norm-bounded region, and two robust optimal design problems are considered. One is aimed at minimising the total transmitter power subject to users' mean square error (MSE) constraints in the presence of channel uncertainty, the other is to minimise the worst-case sum-mean square error (sum-MSE) under power constraints for all admissible uncertainties. For these two problems, the authors propose two iterative algorithms based on second-order cone programming (SOCP) formulations, which can be efficiently solved and have less computational complexity than their semi-definite programming (SDP) counterparts. Simulation results also illustrate that the proposed robust design approaches can significantly reduce the computational complexity while achieving almost the same performance as the robust SDP methods.

Welfare-Maximizing Correlated Equilibria with an Application to Wireless Communication

AbstractThe set of correlated equilibria is convex and contains all Nash equilibria as special cases. Thus, the social welfare-maximizing correlated equilibrium is amenable to convex analysis and offers social welfare that is at least as good as the game's best performing Nash equilibria. We employ robust semidefinite programming (SDP) for computing the social welfare-maximizing correlated equilibria in static polynomial games, giving rise to a dedicated sequential SDP algorithm, the first of this type that can cope with multivariate strategy sets. We apply this algorithm to a wireless communication problem, where two mutually-interfering transmitters and receivers maximize their channel capacities.

A Matrix-Dilation Approach to Robust Semidefinite Programming and Its Error Bound

A Matrix-Dilation Approach to Robust Semidefinite Programming and Its Error Bound

An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.

EXPLOITING SPARSITY IN THE MATRIX-DILATION APPROACH TO ROBUST SEMIDEFINITE PROGRAMMING

A computationally improved approach is proposed for a robust semidefinite programming problem whose constraint is polynomially dependent on uncertain parameters. By exploiting sparsity, the proposed approach gives an approximate problem smaller in size than the matrix-dilation approach formerly proposed by the group of the first author. Here, the sparsity means that the constraint of the given problem has only a small number of nonzero terms when it is expressed as a polynomial of the uncertain parameters. This sparsity is extracted with a special graph called a rectilinear Steiner arborescence, based on which a reduced-size approximate problem is constructed. The quality of the approximation can be evaluated quantitatively. This evaluation shows that the quality can be improved to any level by dividing the parameter region into small subregions.

Robust ℓ 1 performance analysis for linear systems with parametric uncertainties

In this contribution, a computational approach for analysing the robust ℓ∞-gain (or the robust ℓ1 performance) of uncertain linear systems is developed. In particular, the system's state-space matrices may have a rational dependence on structured parametric time-invariant or time-varying uncertainties. The computation is based on robust semi-definite programming and provides a trade-off between accuracy and computational effort. A novel matrix inequality condition to determine the star-norm of discrete-time systems is derived as an auxiliary result.

Robust Semidefinite Programming Problems with Unknown Function Derivatives

Robust Semidefinite Programming Problems with Unknown Function Derivatives

疎性の利用によるロバスト半正定値計画法の効率化

疎性の利用によるロバスト半正定値計画法の効率化

Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness

Robust semidefinite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well established how to systematically construct relaxations on the basis of the full block S-procedure. In general, such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to suggest novel computationally verifiable conditions for when general classes of linear matrix inequality relaxations do not involve any conservatism. If the convex set of uncertainties is finitely generated, we suggest a novel sequence of relaxations which can be proved to be asymptotically exact. Finally, our results are applied to the particularly relevant robustness analysis problem for linear time-invariant dynamical systems affected by uncertainties that are full ellipsoidal or repeated and contained in intersections of disks or circles. This leads to extensions of known results on relaxation exactness for small block structures, including an elementary proof for tightness of standard structured singular value computations for three full complex uncertainty blocks.

ON NUMERICALLY VERIFIABLE EXACTNESS OF MULTIPLIER RELAXATIONS

Robust semi-definite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well-established how to systematically construct relaxations on the basis of the full block S-procedure. In general such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to investigated in how far recently suggested tests for the exactness of such relaxations are indeed numerically verifiable.

Robust semidefinite programming approach to the separability problem

We express the optimization of entanglement witnesses for arbitrary bipartite states in terms of a class of convex optimization problems known as robust semidefinite programs (RSDPs). We propose, using well known properties of RSDPs, several sufficient tests for separability of mixed states. Our results are then generalized to multipartite density operators.

Robust Filtering via Semidefinite Programming with Applications to Target Tracking

In this paper we propose a novel finite-horizon, discrete-time, time-varying filtering method based on the robust semidefinite programming (SDP) technique. The proposed method provides robust performance in the presence of norm-bounded parameter uncertainties in the system model. The robust performance of the proposed method is achieved by minimizing an upper bound on the worst-case varianceof the estimation error for all admissible systems. Our method is recursive and computationally efficient. In our simulations, the new method provides superior performance to some of the existing robust filtering approaches. In particular, when applied to the problem of target tracking, the new method has led to a significant improvement in tracking performance. Our work shows that the robust SDP technique and the interior point algorithms can bring substantial benefits to practically important engineering problems.