Semidefinite Programming Problem Research Articles

A regularized version of the primal-dual Interior Point Method (IPM) for the solution of Semidefinite Programming Problems (SDPs) is presented in this paper. Leveraging on the proximal point method, a novel Proximal Stabilized Interior Point Method for SDP (PS-SDP-IPM) is introduced. The method is strongly supported by theoretical results concerning its convergence: the worst-case complexity result is established for the inner regularized infeasible inexact IPM solver. The new method demonstrates an increased robustness when dealing with problems characterized by ill-conditioning or linear dependence of the constraints without requiring any kind of pre-processing. Extensive numerical experience is reported to illustrate advantages of the proposed method when compared to the state-of-the-art solver.

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results.

Optimal resource allocation for rapid convergence to stable healthy state in epidemic spreading models

This article investigates the problem of inverse optimal control (IOC) for a class of nonlinear affine systems. An adaptive IOC approach is proposed to recover the cost functional using only the system state data, which integrates the finite-time concurrent learning (FTCL) technique and the semidefinite programming (SDP) technique. First, an identifier neural network (NN) is employed to approximate the unknown nonlinear control policy, and an FTCL-based update law is proposed to estimate the weights of the identifier NN online, which removes the traditional persistent excitation (PE) condition. Moreover, the finite-time convergence as well as the uniformly ultimately boundness (UUB) of estimation error of the identifier NN weights are analysed according to whether or not there exists the identifier NN approximation error. Then, with the help of a value NN for approximating the value function, an SDP problem with a quadratic objective function can be set up for determining the weighting matrices of the cost functional. Finally, simulation results are presented to validate the proposed method.

Abstract Tensegrity systems represent promising candidate mechanisms with in-situ stiffness variability through changing the cables' prestress levels. However, prestress-based stiffness behaviors of tensegrity systems with arbitrary kinematic joints have not been analyzed systematically. This paper adopts the natural absolute coordinates for static modeling of tensegrity systems consisting of rigid members and tension elements. Then, a generic stiffness analysis method is developed to formulate the reduced-basis tangent stiffness matrix, which is found to include three parts: positive semi-definite material and geometric stiffness matrices, and an indefinite constraint stiffness matrix. Based on these findings, a systematic stability-checking procedure is derived to determine prestress and super stability, which are qualitative indicators of the softening and stiffening effects in different tensegrity systems. Then, we proceed to quantify the range of prestress-based stiffness variability by formulating semidefinite programming problems that numerically pinpoint the maximum and zero stiffness points. Furthermore, this paper reveals the composable nature of multiple self-stress states, enabling the composability of stiffness properties in mechanism designs. Several numerical examples verify the efficacy and versatility of the proposed method and demonstrate interesting stiffness behaviors of tensegrity systems with kinematic joints.

A topological network connectivity design problem based on spectral analysis

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

Abstract This paper presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad‐hoc selection of split locations. In this work, we eliminate the need for this ad‐hoc selection by introducing an optimization‐based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane’s compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified Föppl’s model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain–displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. The primal variables are displacement functions, whilst in the dual one seeks stresses being Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

Lossless convexification and duality

In this article, we investigate the duality theorems for a class of non-smooth semidefinite multiobjective programming problems with equilibrium constraints (in short, NSMPEC) via convexificators. Utilizing the properties of convexificators, we present Wolfe-type (in short, WMPEC) and Mond-Weir-type (in short, MWMPEC) dual models for the problem NSMPEC. Furthermore, we establish various duality theorems, such as weak, strong, and strict converse duality theorems relating to the primal problem NSMPEC and the corresponding dual models, in terms of convexificators. Numerous illustrative examples are furnished to demonstrate the importance of the established results. Furthermore, we discuss an application of semidefinite multiobjective programming problems in approximating K-means-type clustering problems. To the best of our knowledge, duality results presented in this paper for NSMPEC using convexificators have not been explored before.

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Specific target pinning control of complex dynamical networks based on semidefinite programming strategy

Super-resolution delay-Doppler estimation for OTFS-based automotive radar

This paper addresses the unconstrained binary quadratic problem (UQP). This problem consists in minimizing a quadratic function a binary variables (0−1 variables). Accordingly, in this work, a hybrid algorithm called (HA), based on simulated annealing algorithm with some combination procedures, have been proposed and a branch and bound procedure, based on this algorithm (HA) and semidefinite programming problem (SDP), has been applied. The purpose of this approach is to facilitate the resolution of the initial problem and reduce its dimension by using some fixing criteria in a repeat loop. Numerical results are presented to consolidate the demonstrated theoretical results and prove effectiveness and performance in speed and quality of our new approach.

The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach.

Resilient interval observer-based coordination control for multi-agent systems under stealthy attacks

In this work, we initiate a positive semi-definite numerical bootstrap program for multi-point correlators. Considering six-point functions of operators on a line, we reformulate the crossing symmetry equation for a pair of comb-channel expansions as a semi-definite programming problem. We provide two alternative formulations of this problem. At least one of them turns out to be amenable to numerical implementation. Through a combination of analytical and numerical techniques, we obtain rigorous bounds on CFT data in the triple-twist channel for several examples.

This paper considers filtering for linear systems subjected to persistent exogenous disturbances. The filtering quality is characterized by the size of the bounding ellipsoid that contains the estimated output of the system. A regular approach is proposed to solve the nonfragile filtering problem. This problem consists in designing a filter matrix that withstands admissible variations of its coefficients. The concept of invariant ellipsoids is applied to reformulate the original problem in terms of linear matrix inequalities and reduce it to a parametric semidefinite programming problem easily solved numerically. This paper continues the series of author’s research works devoted to filtering under nonrandom bounded exogenous disturbances and measurement errors.

Abstract A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.