# Positivity and Optimization for Semi-Algebraic Functions

@article{Lasserre2010PositivityAO, title={Positivity and Optimization for Semi-Algebraic Functions}, author={Jean B. Lasserre and Mihai Putinar}, journal={SIAM J. Optim.}, year={2010}, volume={20}, pages={3364-3383} }

We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis on algebras generated by semi-algebraic functions. In this case the standard global optimization problem, with constraints given by elements of the same algebra, is reduced via a natural change of variables to the better-understood case of polynomial optimization. A collection of simple examples and numerical experiments complement the…

## 25 Citations

Positivity and Optimization: Beyond Polynomials

- Mathematics
- 2012

The present chapter offers a glimpse at a series of specific non-polynomial optimization problems, by identifying in every instance the specific results needed to run a robust algebraic relaxation scheme.

An Introduction to Polynomial and Semi-Algebraic Optimization

- Mathematics
- 2015

This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic…

A Striktpositivstellensatz for measurable functions (corrected version)

- Mathematics
- 2009

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of…

Positivstellensätze for real function algebras

- Mathematics
- 2010

We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar [Positivity and…

Formal Proofs for Nonlinear Optimization

- Mathematics, Computer ScienceJ. Formaliz. Reason.
- 2015

The implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates and produces both valid underestimators and lower bounds for each approximated constituent.

Certification of Bounds of Non-linear Functions: The Templates Method

- Mathematics, Computer ScienceMKM/Calculemus/DML
- 2013

An approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis), is introduced, which leads to semialgebraic optimization problems, solved by sum-of-squares relaxations.

Semidefinite programming relaxations for linear semi-infinite polynomial programming

- Mathematics, Computer Science
- 2015

The SDP relaxation method is extended to more general semi-infinite programming problems and it is shown how to verify the compactness of feasible sets of LSIPP problems.

Noncommutative Semialgebraic Sets in Nilpotent Variables

- Mathematics
- 2011

We solve the lifting problem in C^*-algebras for many sets of relations that include the relations x_j^{N_j} = 0 on each variable. The remaining relations must be of the form \| p(x_1,...,x_n) \|…

Semidefinite Approximations of Projections and Polynomial Images of SemiAlgebraic Sets

- Mathematics, Computer ScienceSIAM J. Optim.
- 2015

This work considers the problem of approximating the image set F = f(S) in R^m, and provides a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F.

Certification of inequalities involving transcendental functions: Combining SDP and max-plus approximation

- Mathematics2013 European Control Conference (ECC)
- 2013

A certification method is introduced, combining semialgebraic optimization and max-plus approximation, to certify numerical inequalities used in the proof of the Kepler conjecture by Thomas Hales.

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