On Polynomial Optimization Over Non-compact Semi-algebraic Sets

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

We consider the class of polynomial optimization problems inf{f(x) : x ∈ K} for which the quadratic module generated by the polynomials that define K and the polynomial c − f (for some scalar c) is Archimedean. For such problems, the optimal value can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically. Moreover, the Ar...

Minimizing trigonometric matrix polynomials over semi-algebraic sets

This paper addresses the problem of minimizing the minimum eigenvalue of a trigonometric matrix polynomial. The contribution is to show that, by exploiting Putinar’s Positivstellensatz and introducing suitable transformations, it is possible to derive a nonconservative approach based on semidefinite programming (SDP) whose computational burden can be significantly smaller than that of an existi...

On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities

We consider symmetric (as well as multi-symmetric) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of fixed degrees. We give polynomial (in the dimension of the ambient space) bounds on the number of irreducible representations of the symmetric group which acts on these sets, as well as their multip...

Generating semi-algebraic invariants for non-autonomous polynomial hybrid systems

Hybrid systems are dynamic systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid ...

Polynomial datapath synthesis and optimization based on vanishing polynomial over Z2m and algebraic techniques