Search

In this paper, we focus on the “positive” l 2 induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the “positive” l 2 induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the “positive” l 2 induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.

This paper deals with the minimization of $$H_\infty $$ H ∞ output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed $$H_\infty $$ H ∞ norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular $$H_\infty $$ H ∞ output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction.

We propose a method to reduce the sizes of SDP relaxation problems for a given polynomial optimization problem (POP). This method is an extension of the elimination method for a sparse SOS polynomial in (8) and exploits sparsity of polynomials involved in a given POP. In addition, we show that this method is a partial application of a facial reduction algorithm, which generates a smaller SDP problem with an interior feasible solution. In general, SDP relaxation problems for POPs often become highly degenerate because of a lack of interior feasible solutions. As a result, the resulting SDP relaxation problems obtained by this method may have an interior feasible solution, and one may be able to solve the SDP relaxation problems effectively. Numerical results in this paper show that the resulting SDP relaxation problems obtained by this method can be solved fast and accurately.

The facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial [M. Kojima, S. Kim, H. Waki, Sparsity in sums of squares of polynomials, Math. Program. 103 (2005) 45-62] removes monomials which do not appear in any SOS representations. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial.

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations.

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse polynomials by eliminating redundancy.

Sequences of generalized Lagrangian duals and their sums of squares (SOS) of polynomials relaxations for a polynomial optimization problem (POP) are introduced. The sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the sequence converge to the optimal value of the POP using a method from the penalty function approach. The sequence of SOS relaxations is transformed into a sequence of semidefinite programing (SDP) relaxations of the POP, which correspond to duals of modification and generalization of SDP relaxations given by Lasserre for the POP.