Invariance Conditions for Nonlinear Dynamical Systems

Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation}] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions

Strong duality in conic linear programming: facial reduction and extended duals

The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program $$(P) \sup {<c, x> | Ax \leq_K b}$$ in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana's dual using certificates from a facial reduction algorithm. Here we give a clear and self-contained exposition of facial reduction, of extended duals, and generalize Ramana's dual: -- we state a simple facial reduction algorithm and prove its correctness; and -- building on this algorithm we construct a family of extended duals when $K$ is a {\em nice} cone. This class of cones includes the semidefinite cone and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of a facial reduction algorithm and extended dual systems", technical report, Columbia University, 2000, available from http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of Jonathan Borwein's 60th birthday, 201

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

Matrix pencil, Quadratic programming, Existence theory,

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré’s algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists

Preprocessing and Regularization for Degenerate Semidefinite Programs

This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primal-dual interior-point, p-d i-p methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, well-posedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the Borwein-Wolkowicz facial reduction process to find a finite number, k, of rank-revealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, well-posed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. Th