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… 
View PDF on arXiv
25 Citations
Highly Influential Citations
3
Background Citations
9
Methods Citations
6

25 Citations

Positivity and Optimization: Beyond Polynomials
TLDR
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
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)
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
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
TLDR
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
TLDR
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
  • Feng Guo
  • Mathematics, Computer Science
  • 2015
TLDR
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
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
TLDR
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
TLDR
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.
...
...

References

SHOWING 1-10 OF 43 REFERENCES
Positive Polynomials on Semialgebraic Sets
In this chapter we study improvements of Theorem 5.2.9, which, for polynomials strictly positive on a bounded semialgebraic set of the form W ℝ(h 1 ,..., h s ), gives a canonical representation…
Emerging applications of algebraic geometry
Polynomial Optimization on Odd-Dimensional Spheres.- Engineering Systems and Free Semi-Algebraic Geometry.- Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood…
Real algebraic geometry
1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem.…
The Moment Problem on Compact Semi-Algebraic Sets
In this chapter we begin the study of the multidimensional moment problem. The passage to dimensions d ≥ 2 brings new difficulties and unexpected phenomena. In Sect. 3.2 we derived solvability…
Semidefinite programming relaxations for semialgebraic problems
  • P. Parrilo
  • Mathematics, Computer Science
    Math. Program.
  • 2003
TLDR
It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz.
POSITIVE POLYNOMIALS IN SCALAR AND MATRIX VARIABLES, THE SPECTRAL THEOREM AND OPTIMIZATION
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decomposi- tions, with emphasis on the interaction between classical moment…
Global Optimization with Polynomials and the Problem of Moments
TLDR
It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.
Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.
Suppose R is a real closed field, with the usual, order topology; let K be a subfield, with the inherited order. Let X--= {X19 ..., Xn} be indeterminates. We call feK(X"] positive semidefinite (psd)…
Gaussian Interval Quadrature Formulae for Tchebycheff Systems
TLDR
It is proved the existence and uniqueness of the Gaussian interval quadrature formula based on n weighted integrals over nonoverlapping subintervals of [a,b] of preassigned lengths.
Optimal Inequalities in Probability Theory: A Convex Optimization Approach
TLDR
It is shown that it is NP-hard to find tight bounds for k = 2 and $\Omega={\cal R}^n$, and an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.
...
...