Results 1  10
of
35
Semidefinite representation of convex sets
, 2007
"... Let S = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is compact, convex and has nonempty interior. Let Si = {x ∈ R n: gi(x) ≥ 0} and ∂Si = {x ∈ R n: gi(x) = 0} be its boundary. This paper, as does the subject of semidefin ..."
Abstract

Cited by 47 (10 self)
 Add to MetaCart
Let S = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is compact, convex and has nonempty interior. Let Si = {x ∈ R n: gi(x) ≥ 0} and ∂Si = {x ∈ R n: gi(x) = 0} be its boundary. This paper, as does the subject of semidefinite programming (SDP), concerns Linear Matrix Inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable [6]. A question arising from [13], see [6, 14], is: given S ∈ R n, does there exist an LMI representable set ˆ S in some higher dimensional space R n+N whose projection down onto R n equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) Assume gi(x) are all concave on S. If the positive definite Lagrange Hessian (PDLH) condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each gi(x) is either sosconcave (− ∇ 2 gi(x) = W(x) T W(x) for some matrix polynomial W(x)) or strictly quasiconcave on S, then S is SDP representable. (iii) If each Si is either sosconvex or poscurvconvex (Si is compact, convex and has smooth boundary with positive curvature), then S is SDP representable. This also holds for Si for which ∂Si ∩ S extends smoothly to the boundary of a poscurvconvex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)(iii).
Selected topics in robust convex optimization
 MATH. PROG. B, THIS ISSUE
, 2007
"... Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robu ..."
Abstract

Cited by 32 (2 self)
 Add to MetaCart
Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.
LMI techniques for optimization over polynomials in control: a survey
 IEEE Transactions on Automatic Control
"... Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the ..."
Abstract

Cited by 31 (17 self)
 Add to MetaCart
(Show Context)
Abstract—Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya’s theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, timedelay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.
PURE STATES, POSITIVE MATRIX POLYNOMIALS AND SUMS OF HERMITIAN SQUARES
, 2009
"... Let M be an archimedean quadratic module of real t×t matrix polynomials in n variables, and let S ⊆ R n be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P t−1 (R). This leads us to conceptua ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Let M be an archimedean quadratic module of real t×t matrix polynomials in n variables, and let S ⊆ R n be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P t−1 (R). This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for nonsymmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
Stability and Stabilization of Aperiodic SampledData Control Systems Using Robust Linear Matrix Inequalities
, 2009
"... Stability analysis of an aperiodic sampleddata control system is discussed for application to network and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a t ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Stability analysis of an aperiodic sampleddata control system is discussed for application to network and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient condition can be constructed with the mean value theorem. Special care is paid on tightness of the sufficient condition for less conservative stability analysis. Asymptotic exactness of the approach is discussed and a technique of adaptive division is presented for computational efficiency. Extension to stabilization is also discussed. Examples show the efficacy of the approach.
A Quantitative Pólya’s Theorem with Zeros
, 2007
"... Let R[X]: = R[X1,..., Xn] and let R + [X] denote polynomials in R[X] with nonnegative coefficients. Pólya’s Theorem [5] says that if p is a homogeneous polynomial in n variables ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Let R[X]: = R[X1,..., Xn] and let R + [X] denote polynomials in R[X] with nonnegative coefficients. Pólya’s Theorem [5] says that if p is a homogeneous polynomial in n variables
LMI tests for positive definite polynomials: Slack variable approach
 IEEE Trans. on Automatic Control
"... The considered problem is assessing nonnegativity of a function’s values when indeterminates are in domains constrained by scalar polynomial inequalities. The tested functions are multiindeterminates polynomial matrices which are required to be positive semidefinite. For such problems new tests b ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The considered problem is assessing nonnegativity of a function’s values when indeterminates are in domains constrained by scalar polynomial inequalities. The tested functions are multiindeterminates polynomial matrices which are required to be positive semidefinite. For such problems new tests based on linear matrix inequalities are provided in a Slack Variables type approach. The results are compared to those obtained via the SumOfSquares approach, are proved to be equivalent in case of unbounded domains and less conservative if polytopictype bounds are known.
Infeasibility certificates for linear matrix inequalities
"... Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebr ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality L(x) ≽ 0 is infeasible if and only if −1 lies in the quadratic module associated to L. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certificate, we use a more involved algebraic certificate motivated by the real radical and Prestel’s theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination. A linear matrix inequality (LMI) is a condition of the form n∑ L(x) = A0 + xiAi ≽ 0 (x ∈ R n)