Linear Semidefinite Programming Research Articles

Loraine – an interior-point solver for low-rank semidefinite programming

The aim of this paper is to introduce a new code for the solution of large-and-sparse linear semidefinite programs (SDPs) with low-rank solutions or solutions with few outlying eigenvalues, and/or problems with low-rank data. We propose to use a preconditioned conjugate gradient method within an interior-point SDP algorithm and an efficient preconditioner fully utilizing the low-rank information. The efficiency is demonstrated by numerical experiments using the truss topology optimization problems, Lasserre relaxations of the MAXCUT problems and the sensor network localization problems.

Logarithmic Barrier Method Via Minorant Function for Linear Semidefinite Programming

Abstract We propose in this study, a new logarithmic barrier approach to solve linear semidefinite programming problem. We are interested in computation of the direction by Newton’s method and of the displacement step using minorant functions instead of line search methods in order to reduce the computation cost. Our new approach is even more beneficial than classical line search methods. This purpose is confirmed by some numerical simulations showing the e˙ectiveness of the algorithm developed in this work, which are presented in the last section of this paper.

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.

A smoothing gradient-based neural network strategy for solving semidefinite programming problems

ABSTRACT Linear semidefinite programming problems have received a lot of attentions because of large variety of applications. This paper deals with a smooth gradient neural network scheme for solving semidefinite programming problems. According to some properties of convex analysis and using a merit function in matrix form, a neural network model is constructed. It is shown that the proposed neural network is asymptotically stable and converges to an exact optimal solution of the semidefinite programming problem. Numerical simulations are given to show that the numerical behaviours are in good agreement with the theoretical results.

Sums of Squares Polynomial Program Reformulations for Adjustable Robust Linear Optimization Problems with Separable Polynomial Decision Rules

We show that adjustable robust linear programs with affinely adjustable box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) polynomial reformulations. These problems share the same optimal values and admit a one-to-one correspondence between the optimal solutions. A sum of squares representation of non-negativity of a separable non-convex polynomial over a box plays a key role in the reformulation. This reformulation allows us to find adjustable robust solutions of uncertain linear programs under box data uncertainty by numerically solving their associated equivalent SOS polynomial optimization problem using semi-definite linear programming. We illustrate how the quality of the adjustable robust solution of a robust optimization problem with polynomial decision rules improves as the degree of the polynomial increases. Our results demonstrate that the adjustable robust solutions approach the actual optimal solution as the degree of the polynomial increases from one to fifteen.

Uniform exponential stability criteria with verification for nonautonomous switched nonlinear systems with uncertainty

AbstractThis article is concerned with the sufficient and necessary conditions for the global uniform exponential stability (GUES) of a class of uncertain nonautonomous switched nonlinear systems (UNSNS), that is, systems with time‐varying uncertain information. To start with, based on a piecewise continuous scalar function and a piecewise differentiable uncertainty‐dependent Lyapunov function along the trajectories, we propose alternative sufficient and necessary conditions for the GUES of UNSNS. Afterwards, for removing the dependence on uncertain information of the required Lyapunov functions, we combine an uncertainty‐independent Lyapunov function with its right upper Dini derivative and the above piecewise continuous scalar function to further obtain other alternative sufficient and necessary conditions for the GUES of UNSNS. Note that our conditions release the requirement on negative definiteness of the time‐derivatives and Dini derivatives of Lyapunov functions respectively, which is illustrated by Example 1. Further, for the rational UNSNS, we propose a linear semidefinite programming based computable approach to mechanically verify our current theoretical results. In the end, the effectiveness of our theoretical results and mechanical approach are illustrated by two examples.

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Iterative Process

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Iterative Process

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Newton’s System of Equations

Primal–Dual Newton Method with Steepest Descent for the Linear Semidefinite Programming Problem: Newton’s System of Equations

Distributionally Robust State Estimation for Linear Systems Subject to Uncertainty and Outlier

Parameter uncertainties and measurement outliers unavoidably exist in a real linear system. Such uncertainties and outliers make the true joint state-measurement distributions (induced by the true system model) deviate from the nominal ones (induced by the nominal system model) so that the performance of the optimal state estimator designed for the nominal model becomes unsatisfactory or even unacceptable in practice. The challenges are to quantitatively describe the uncertainties in the model and the outliers in the measurements, and then robustify the estimator in a right way. This article studies a distributionally robust state estimation framework for linear systems subject to parameter uncertainties and measurement outliers. It utilizes a family of distributions near the nominal one to implicitly describe the uncertainties and outliers, and the robust state estimation in the worst case is made over the least-favorable distribution. The advantages of the presented framework include: 1) it only uses a few scalars to parameterize the method and does not require the structural information of uncertainties; 2) it generalizes several classical filters (e.g., the fading Kalman filter, risk-sensitive Kalman filter, relative-entropy Kalman filter, outlier-insensitive Kalman filters) into a unified framework. We show that the distributionally robust state estimation problem can be reformulated into a linear semi-definite program and in some special cases it can be analytically solved. Comprehensive comparisons with existing state estimation frameworks that are insensitive to parameter uncertainties and measurement outliers are also conducted.

Theoretical and numerical result for linear semidefinite programming based on a new kernel function

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

Calculating Radius of Robust Feasibility of Uncertain Linear Conic Programs via Semi-definite Programs

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise, the robust counterpart of an uncertain program is not well defined as a robust optimization problem. In this paper, we address a key fundamental question of robust optimization: How to compute the radius of robust feasibility of uncertain linear conic programs, including linear programs? We first provide computable lower and upper bounds for the radius of robust feasibility for general uncertain linear conic programs under the commonly used ball uncertainty set. We then provide important classes of linear conic programs where the bounds are calculated by finding the optimal values of related semi-definite linear programs, among them uncertain semi-definite programs, uncertain second-order cone programs and uncertain support vector machine problems. In the case of an uncertain linear program, the exact formula allows us to calculate the radius by finding the optimal value of an associated second-order cone program.

Angle-Based Sensor Network Localization

This article studies angle-based sensor network localization (ASNL) in a plane, which is to determine locations of all sensors in a sensor network, given locations of partial sensors (called anchors) and angle measurements obtained in the local coordinate frame of each sensor. First, it is shown that a framework with a nondegenerate bilateration ordering must be angle fixable, implying that it can be uniquely determined by angles between edges up to translations, rotations, reflections, and uniform scaling. Then, ASNL is proved to have a unique solution if and only if the grounded framework is angle fixable and anchors are not all collinear. Subsequently, ASNL is solved in centralized and distributed settings, respectively. The centralized ASNL is formulated as a rank-constrained semidefinite program (SDP) in either a noise-free or a noisy scenario, with a decomposition approach proposed to deal with large-scale ASNL. The distributed protocol for ASNL is designed based on intersensor communications. Graphical conditions for equivalence of the formulated rank-constrained SDP and a linear SDP, decomposition of the SDP, as well as the effectiveness of the distributed protocol, are proposed, respectively. Finally, simulation examples demonstrate our theoretical results.

Analysis and Verification of Input-to-State Stability for Nonautonomous Discrete-Time Switched Systems via Semidefinite Programming

This article concerns the theoretical analysis and mechanical verification of input-to-state stability (ISS) for nonautonomous discrete-time switched systems. To start with, based on a bounded function and the average dwell time, we successively propose less conservative sufficient conditions for uniform input-to-state stability, global uniform asymptotic input-to-state stability, and global uniform exponential input-to-state stability of nonautonomous switched nonlinear systems. Then, for systems with zero inputs, we apply our bounded function and average dwell time based method to further relax the sufficient conditions for their uniform stability, global uniform asymptotic stability, and global uniform exponential stability. Particularly, we propose a linear semidefinite programming based computable approach for mechanical verification of our current theoretical results for the rational (and even certain nonrational) nonautonomous switched systems. Note that our theoretical results and mechanical approach are both illustrated by examples.

Linear semidefinite programming problems: regularisation and strong dual formulations

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

Strengthening convex relaxations of 0/1-sets using Boolean formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S subseteq {0,1}^n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula phi defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula phi . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

Inner-Estimating Domains of Attraction for Nonpolynomial Systems With Polynomial Differential Inclusions.

In this article, based on polynomial differential inclusions, we propose a heuristic iterative approach for estimating the domains of attraction for nonpolynomial systems. First, we use the fuzzy model to construct a polynomial differential inclusion for the nonpolynomial system, which can be equivalently written as a time-invariant uncertain polynomial system. Then, beginning with an initial inner estimation, we present an iterative approach to enlarge this initial inner estimation by calculating common Lyapunov-like functions. Furthermore, the domains of attraction are estimated by combining this iterative approach with heuristic construction of differential inclusions. In the end, our heuristic iterative approach is implemented with linear semidefinite programming and then tested on some nonpolynomial examples with comparisons to the existing methods in the literature.

Counting frequent patterns in large labeled graphs: a hypergraph-based approach.

In recent years, the popularity of graph databases has grown rapidly. This paper focuses on single-graph as an effective model to represent information and its related graph mining techniques. In frequent pattern mining in a single-graph setting, there are two main problems: support measure and search scheme. In this paper, we propose a novel framework for designing support measures that brings together existing minimum-image-based and overlap-graph-based support measures. Our framework is built on the concept of occurrence/instance hypergraphs. Based on such, we are able to design a series of new support measures: minimum instance (MI) measure, and minimum vertex cover (MVC) measure, that combine the advantages of existing measures. More importantly, we show that the existing minimum-image-based support measure is an upper bound of the MI measure, which is also linear-time computable and results in counts that are close to number of instances of a pattern. We show that not only most major existing support measures and new measures proposed in this paper can be mapped into the new framework, but also they occupy different locations of the frequency spectrum. By taking advantage of the new framework, we discover that MVC can be approximated to a constant factor (in terms of number of pattern nodes) in polynomial time. In contrast to common belief, we demonstrate that the state-of-the-art overlap-graph-based maximum independent set (MIS) measure also has constant approximation algorithms. We further show that using standard linear programming and semidefinite programming techniques, polynomial-time relaxations for both MVC and MIS measures can be developed and their counts stand between MVC and MIS. In addition, we point out that MVC, MIS, and their relaxations are bounded within constant factor. In summary, all major support measures are unified in the new hypergraph-based framework which helps reveal their bounding relations and hardness properties.

ON OPTIMUM DESIGN OF FRAME STRUCTURES

Optimization of frame structures is formulated as a non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.

A robust bank asset allocation model integrating credit-rating migration risk and capital adequacy ratio regulations

In this paper, we consider a bank asset allocation problem with uncertain migration risk of credit ratings and capital adequacy ratio (CAR) regulations. In the practical scenarios, the future market values of each risky asset are largely affected by outer complex environments. We only observe the information about their first-moment and marginal second-moment of year-ahead market value of each loan asset. Based on these scenarios, we propose a new distributionally robust optimization model with the chance constraint characterized by uncertain CAR. Following the duality theory in infinite-dimensional optimization problem and the theory of conic linear optimization model, we can reformulate the original problem into a tractable linear deterministic semi-definite programming (SDP) model. By using this tractable linear SDP model, we can provide a robust asset allocation policy to bank managers. Further, we conduct a simulation study to illustrate the application of our method under two different economic conditions, a downward condition and an upward condition. Then a series of sensitivity tests is applied to examine the impacts of various factors, including safety probability, target CAR and recovery rate, on the optimal asset allocations. We also compare the performance of our model and the CVaR model. We demonstrate our model provides an efficient way to deal with the trade-off between expected return and CAR.

Quantitative stability of two-stage distributionally robust risk optimization problem with full random linear semi-definite recourse

In this paper, we study a distributionally robust risk optimization (DRRO) problem where the information on the probability distribution of the underlying random variables is incomplete. But it is possible to use partial information to construct an ambiguity set of probability distributions. In some cases, decision vector x may have a direct impact on the likelihood of the underlying random events that occur after the decision is taken, which motivates us to propose an ambiguity set to be parametric and decision-dependent. To conduct quantitative stability analysis of the optimal value function and the optimal solution mapping of the DRRO problem, we derive error bounds results for the parametrized ambiguity set under the total variation metric and investigate Lipschitz continuity of the objective function of the DRRO problem under some conditions. As an application, we demonstrate that the two-stage stochastic linear semi-definite programs satisfy these conditions and then apply results obtained to it.