Optimality conditions and finite convergence of Lasserre’s hierarchy

The optimal value of a polynomial optimization over a compact semialgebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semialgebraic feasible sets, for which the associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. We show that for such problems, the corresponding hierachy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.

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“…In addition, as recently proved by Nie [7] and Marshall [8], finite convergence is generic for POPs on compact basic semi-algebraic sets.…”

Section: Introductionmentioning

confidence: 55%

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

Jeyakumar

Lasserre

2014

J Optim Theory Appl

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“…This is in contrast with the semidefinite relaxations (1.3) for which finite convergence takes place for convex problems where ∇ 2 f (x * ) is positive definite at every minimizer x * ∈ K (see de Klerk and Laurent [6,Corollary 3.3]) and occurs at the first relaxation for SOS-convex 1 problems [11,Theorem 3.3]. In fact, as demonstrated in recent works of Marshall [14] and Nie [15], finite convergence is generic (even for non convex problems).…”

mentioning

confidence: 71%

“…And in contrast to LP-relaxations, there is no obstruction to exactness (i.e., finite convergence). In fact, it is quite the opposite since as demonstrated recently in Nie [15], finite convergence is generic!…”

Section: On the Hierarchy Of Semidefinite Relaxationsmentioning

confidence: 95%

A Lagrangian Relaxation View of Linear and Semidefinite Hierarchies

Lasserre¹

2013

SIAM J. Optim.

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We consider the general polynomial optimization problem P : f * = min{f (x) : x ∈ K} where K is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum f * , provided that it is applied to a problemP equivalent to P, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of P. Next we show that the standard hierarchy of LP-relaxations of P (in the spirit of Sherali-Adams' RLT) can be interpreted as a brute force simplification of the above Lagrangian relaxation in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, we provide a systematic improvement of the LP-hierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not linear programs any more). For each semidefinite program in the parametrized hierarchy, the semidefinite constraint has a fixed size O(n k ), independently of the rank in the hierarchy, in contrast with the standard hierarchy of semidefinite relaxations. The parameter k is to be decided by the user. When applied to a non trivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LP-hierarchy where convergence cannot be finite. When applied to 0/1 programs it is at least as good as the first one in the hierarchy of semidefinite relaxations. However obstructions to exactness still exist and are briefly analyzed. Finally, the standard semidefinite hierarchy can also be viewed as a simplification of an extended Lagrangian relaxation, but different in spirit as sums of squares (and not scalars) multipliers are allowed.

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“…In fact, by results from Marshall [13] and more recently Nie [14], membership in Q(g) is also generic for polynomials that are only nonnegative on K. And so Putinar's Positivstellensatz is particularly useful to certify and enforce that a polynomial is nonnegative on K, and in particular the polynomial x → f (x) − f (y) for the inverse optimization problem associated with a feasible solution y ∈ K.…”

Section: Introductionmentioning

confidence: 99%

Inverse polynomial optimization

Lasserre

2015

An Introduction to Polynomial and Semi-Algebraic Optimization

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Abstract. We consider the inverse optimization problem associated with the polynomial program f * = min{f (x) : x ∈ K} and a given current feasible solution y ∈ K. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomialf (which may be of same degree as f if desired) with the following properties: (a) y is a global minimizer off on K with a Putinar's certificate with an a priori degree bound d fixed, and (b),f minimizes f −f (which can be the ℓ 1 , ℓ 2 or ℓ∞-norm of the coefficients) over all polynomials with such properties. Computingf d reduces to solving a semidefinite program whose optimal value also provides a bound on how far is f (y) from the unknown optimal value f * . The size of the semidefinite program can be adapted to computational capabilities available. Moreover, if one uses the ℓ 1 -norm, thenf takes a simple and explicit canonical form. Some variations are also discussed.

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Optimality conditions and finite convergence of Lasserre’s hierarchy

Cited by 218 publications

References 24 publications

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

A Lagrangian Relaxation View of Linear and Semidefinite Hierarchies

Inverse polynomial optimization

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