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210
Global Optimization with Polynomials and the Problem of Moments
 SIAM Journal on Optimization
, 2001
"... We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear mat ..."
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Cited by 569 (47 self)
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We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22, 90C25 PII. S1052623400366802 1.
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The mai ..."
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Cited by 361 (22 self)
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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SOSTOOLS: Sum of squares optimization toolbox for MATLAB
, 2004
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Sums of squares of regular functions on real algebraic varieties
 Tran. Amer. Math. Soc
, 1999
"... Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a ..."
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Cited by 66 (16 self)
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Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth nonrational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact ” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counterexamples of genus 1.
Optimization of polynomials on compact semialgebraic sets
 SIAM J. Optim
"... Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter conve ..."
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Cited by 59 (4 self)
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Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ∗ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x ∗ , we prove that every sequence of “nearly ” optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x ∗. 1. Introduction to Lasserre’s method Throughout the paper, we suppose 1 ≤ n ∈ N and abbreviate (X1,..., Xn) by ¯X. We let R [ ¯ X] denote the ring of real polynomials in n indeterminates. Suppose we are given a so called basic closed semialgebraic set, i.e., a set S: = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0}
The truncated complex Kmoment problem
 TRANS. AMER. MATH. SOC
, 2000
"... Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there exists a positive Borel measure µ on C such that γij = ..."
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Cited by 59 (12 self)
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Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there exists a positive Borel measure µ on C such that γij = ∫ ¯z izj dµ (0 ≤ i + j ≤ 2n) and supp µ ⊆ K. For K ≡ KP a semialgebraic set determined by a collection of complex polynomials P = {pi (z, ¯z)} m i=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n)(γ) and the localizing matrices Mp i. We prove that there exists a rank M (n)atomic representing measure for γ (2n) supported in KP if and only if M (n) ≥ 0andthereissomerankpreserving extension M (n +1)forwhichMp i (n + ki) ≥ 0, where deg pi =2ki or 2ki − 1(1 ≤ i ≤ m).
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 52 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
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 ..."
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Cited by 49 (10 self)
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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).
Nonlinear optimal control via occupation measures and LMI relaxations
 SIAM Journal on Control and Optimization
, 2008
"... Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state an ..."
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Cited by 47 (24 self)
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Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments. 1.