221 research outputs found
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
Stability and Performance Verification of Optimization-based Controllers
This paper presents a method to verify closed-loop properties of
optimization-based controllers for deterministic and stochastic constrained
polynomial discrete-time dynamical systems. The closed-loop properties amenable
to the proposed technique include global and local stability, performance with
respect to a given cost function (both in a deterministic and stochastic
setting) and the gain. The method applies to a wide range of
practical control problems: For instance, a dynamical controller (e.g., a PID)
plus input saturation, model predictive control with state estimation, inexact
model and soft constraints, or a general optimization-based controller where
the underlying problem is solved with a fixed number of iterations of a
first-order method are all amenable to the proposed approach.
The approach is based on the observation that the control input generated by
an optimization-based controller satisfies the associated Karush-Kuhn-Tucker
(KKT) conditions which, provided all data is polynomial, are a system of
polynomial equalities and inequalities. The closed-loop properties can then be
analyzed using sum-of-squares (SOS) programming
Practical polynomial optimization through positivity certificates with and without denominators
Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant , sous une certaine condition sur légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur , et donc cette description alternative fournit un certificat de positivité sur . De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set can be written as a linear weighted combination of the polynomials describing , under a certain condition on slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on , and therefore this alternative description provides a certificate of positivity on . Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators
Rational Dual Certificates for Weighted Sums-of-Squares Polynomials with Boundable Bit Size
In (Davis and Papp, 2022), the authors introduced the concept of dual
certificates of sum-of-squares polynomials, which are vectors from the dual
cone of the cone of weighted sums of squares (WSOS) polynomials that can be
interpreted as WSOS nonnegativity certificates. This initial theoretical work
showed that for every polynomial in the interior of a WSOS cone, there exists a
rational dual certificate proving that the polynomial is WSOS. In this article,
we analyze the complexity of rational dual certificates of WSOS polynomials by
bounding the bit sizes of integer dual certificates as a function of parameters
such as the degree and the number of variables of the polynomials, or their
distance from the boundary of the cone. After providing a general bound, we
explore a number of special cases, such as univariate polynomials nonnegative
over the real line or a bounded interval, represented in different commonly
used bases. We also provide an algorithm which runs in rational arithmetic and
computes a rational certificate with boundable bit size for a WSOS lower bound
of the input polynomial.Comment: Submitted for publicatio
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