Estimation of univariate regression functions from bounded i.i.d. data is considered. Estimates are deened by minimizing a complexity penalized residual sum of squares over all piecewise polynomials. The integrated squared error of these estimates achieves for piecewise p-smooth regression functions the rate (ln 2 (n)=n) 2p 2p+1 .

This paper presents a descriptor representation-based guaranteed cost design methodology for polynomial fuzzy systems. applies the representation presenting closed-loop system of model with parallel distributed compensation (PDC) based controller. By utility representation, control (GCC) analysis can utilize slack matrices obtaining less conservative results. The proposed GCC is presented as su...

Given a random n-variate degree-d homogeneous polynomial f , we study the following problem:

To prove that a polynomial is nonnegative on R one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods (see, e.g., [Par03]), thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact...

We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms unconstrained convex minimization. The low-cost iteration complexity enjoyed by renders them particularly relevant applications in machine learning large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we hierarchy semidefinite programs that gi...

A constraint satisfaction problem (CSP) is a decision problem for which the instances are sets of constraints, and the answer is ”yes” (the CSP is satisfiable) if there exists an object satisfying all the constraints, and ”no” (the CSP is unsatisfiable) if no such object exists. Examples of CSPs include 3-SAT, graph coloring, and linear programming. A random instance of a CSP is one created by ...

To prove that a polynomial is nonnegative on R one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods (see, e.g., [Par03]), thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact...

This article presents the $L_1$ -gain polynomial fuzzy output-feedback controller design and stability analysis using sum-of-squares (SOS) approach for positive fuzzy-model-based (PPFMB) control systems. The polynomials, positivity, opt...

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are ob...

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are ob...