# Haut PDF Some approximation schemes in polynomial optimization ### Some approximation schemes in polynomial optimization

Apart from these particular cases, the construction of the optimal design relies on numerical optimization procedures. The case of the determinant—which corresponds to the choice q = 0, i.e., the D-optimality—is studied for example in [Wyn70] and [VBW98]. Another criterion based on matrix conditioning—referred to as G-optimality—is developed in [MYZ15]. In the latter paper, the construction of an optimal design is performed in two steps. In the ﬁrst step one only deals with an optimization problem on the set of all possible information matrices, while in the second step, one wishes to identify a possible probability distribution associated with the optimal information matrix. General optimization algorithms are discussed in [Fed10] and [ADT07]. A general optimization frame on measure sets including gradient descent methods is considered in [MZ04]. In the frame of ﬁxed given support points, eﬃcient SDP based algorithms are proposed and studied in [Sag11] and [SH15]. Let us mention the paper [VBW98] which is one of the original motivations to develop SDP solvers, especially for Max-Det-Problems—corresponding to D-optimal design—and the so-called problem of analytical centering.
En savoir plus ### Approximating min-max (regret) versions of some polynomial problems

Keywords: min-max, min-max regret, approximation, fptas, shortest path, min- imum spanning tree. 1 Introduction The definition of an instance of a combinatorial optimization problem requires to specify parameters, in particular objective function coefficients, which may be uncertain or imprecise. Uncertainty/imprecision can be structured through the concept of scenario which corresponds to an assignment of plausible values to parameters. There exist two natural ways of describing the set of all possible scenarios. In the interval data case, each numerical parameter can take any value between a lower and an upper bound. In the discrete scenario case, which is considered here, the scenario set is described explicitly. Kouvelis and Yu  proposed the min-max and min-max regret criteria, stemming from decision theory, to construct solutions hedging against parameters variations. The min- max criterion aims at constructing solutions having a good performance in the worst case. The min-max regret criterion, less conservative, aims at obtaining a solution minimizing the maximum deviation, over all possible scenarios, of the value of the solution from the optimal value of the corresponding scenario.
En savoir plus ### On inhomogeneous diophantine approximation with some quasi-periodic expressions, II

M(e’/-’, 1/2) = 1/8 and M(eI/8, 1/3) =0if~=2 (mod 3); 1/18 otherwise. Then, what is the condition such that ,Jt~l (9, ~) - 0 holds? It seems that a non-constant polynomial in a quasi-periodic part influences whether M (0, Ø) = 0 or not. So, we consider the cases each quasi-periodic form ### Exponential approximation schemata for some network design problems

major breakthrough has been obtained in  where the Hamiltonian cycle problem is solved in O ∗ (1.66 n ). Dealing with polynomial space, the best running time known so far for traveling salesman problem is O ∗ (4 n n log n ) reached by the algorithm in  (see ). In this article, we study in Section 2 the possibility to get ratios arbitrary close to 1 with a running time better than the one of exact computation. We handle min steiner tree and some versions of traveling salesman problem . In both cases, the basic idea is to find a small part of the instance verifying some suitable properties, then to solve the instance on the remaining part and to build finally a global solution. In Section 3, we show how one can take advantage of the possible existence of some polytime r-approximation algorithm in order to reach interesting (though exponential) running times for ratios slightly better than r.
En savoir plus ### Sparse polynomial approximation of parametric elliptic PDEs Part II: lognormal coefficients *

the ` p summability of (j β kψ j k L ∞ ) j≥1 for some β > 1 2 , which still represents an improvement over the condition in . We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen-Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (ku ν k V ) ν∈F . While we focus on the diffusion equation, our analysis applies to
En savoir plus ### Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

VIII. C ONCLUSIONS This article addressed the stability analysis for a class of complementarity systems using the method of Lyapunov functions. Questions pertaining to the existence of cone-copositive Lyapunov functions were answered in the affirmative for exponentially stable systems. Some refinements of this result, under certain conditions on the vector field in the system dynamics, allow us to restrict our search for cone-copositive Lyapunov functions within the class of homogeneous and rational polynomials. These statements indeed bring tractability to the numerical methods that have been proposed in this paper for computing Lyapunov functions. In particular, two hierarchies of convex optimization problems are obtained using the methods based on discretization and SOS approximation, respectively. Several immediate questions of interest emerge from our work which require further investigation. The first one among those is to extend our results to broader classes of complementarity systems. Systems of the form (C-Sys) are one particular class of relative degree one systems, but in applications, one sees more complex complementarity systems of the form studied in . In such a wider class of systems, one sees different kinds of constraints on the state trajectories. Moreover, the constraints may vary with time in which case one has to consider the possibility of time-varying Lyapunov functions. It would be interesting to consider converse questions for this broader class of systems.
En savoir plus ### Approximation algorithms for some vehicle routing problems

a k+1 k − ε standard approximation for Min-Max b n k cTSP(1,2) could decide (k − 4)PP in polynomial time. 6 Min-Sum EkTSP Bellmore and Hong  showed that when the constraint p = k is replaced by p ≤ k, then Min-Sum kTSP is standard equivalent to TSP on an extended graph. This is true even for the directed version of the problem and when there is a cost associated with activating a salesman. For our case the transformation simply involves replacing the depot vertex 0 by k vertices of zero distance. Also, the metric case of the p ≤ k version is not of interest since the solution is just a single cycle (thus, we deal with the case p = k and Min-Sum EkTSP denote this problem).
En savoir plus ### An overview on polynomial approximation of NP-hard problems

paschos@lamsade.dauphine.fr October 29, 2007 Abstract The fact that it is very unlikely that a polynomial time algorithm could ever be devised for optimally solving NP-hard problems, strongly motivates both researchers and practitioners in trying to heuristically solving such problems, by making a trade-off between computational time and solution’s quality. In other words, heuristic computation consists of trying to find in reasonable time, not the best solution but one solution which is “near” the optimal one. Among classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in polynomial time by computing feasible solutions that are, under some predefined criterion, as near as possible to the optimal ones. The polynomial approximation theory deals with the study of such algorithms. This survey presents and analyzes in a first time approximation algorithms for some classical examples of NP -hard problems. In a second time, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what it is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
En savoir plus ### On some difficulties with a posterior probability approximation technique

Although both approximations ˜ ̺ k (y) and ˆ ̺ k (y) differ in their expressions, they fundamentally relate to the same notion that parameters from other models can be ignored when conditioning on the model index M . This approach is therefore bypassing the simultaneous exploration of several parameters spaces and restricts the simulation to marginal samplers on each separate model. This feature is very appealing since it cuts most of the complexity from the schemes both of Carlin and Chib (1995) and of Green (1995). We however question the foundations of those approximations as presented in both Scott (2002) and Congdon (2006, 2007) and advance below arguments that both authors are using incompatible versions of joint distributions on the collection of parameters that jeopardise the validity of the approximations.
En savoir plus ### Numerical approximation of BSDEs using local polynomial drivers and branching processes

Keywords: Bsde, Monte-Carlo methods, branching process. MSC2010: Primary 65C05, 60J60; Secondary 60J85, 60H35. 1 Introduction Since the seminal paper of Pardoux and Peng , the theory of Backward Stochastic Differential Equations (BSDEs hereafter) has been largely developed, and has lead to many applications in optimal control, finance, etc. (see e.g. El Karoui, Peng and Quenez ). Different approaches have been proposed during the last decade to solve them numerically, without relying on pure PDE based resolution methods. A first family of numerical schemes, based on a time discretization technique, has been introduced by Bally and Pagès ,
En savoir plus ### Numerical approximation of parabolic problems by means of residual distribution schemes

The numerical approximation of (1) has already been considered by sev- eral authors. It is not, however, a completely trivial matter. Indeed, this class of schemes has originally been devised for (steady) transport problems, based on a genuinely multidimensional upwind approach. Among the first contribu- tions, one may quote the work of P.L. Roe . Some connections with more classical schemes, as the streamline diffusion method by Hughes, Johnson and co-authors, have soon been made . However, the main problem is that, even though some residual distribution schemes can be recast as a particular class stabilized finite elements with emphasis on L ∞ stability, there is no clear general framework allowing to choose the test functions in order to recover a traditional variational statement. The main reason of this problem is related to the underlying formulation: everything is seen from a discrete point of view, and emphasis is put on the point-wise behavior of the residual (which is natural given the focus on L ∞ stability). Indeed the same remark applies to variants of the method not aiming at approximating point values of the solution, as in . In this case, the local discrete (point-wise) residuals are replaced by residuals for polynomial coefficient sets for which once again a maximum principle is sought for.
En savoir plus ### Polynomial time approximation schemes for dense instances of minimum constraint satisfaction

a factor n Ω(1)/ log log n (cf. [KST97], [DKS98], [DKRS00]), and this hardness ratio is in fact also valid for average dense instances. Only recently a polynomial time algorithm with the first sublinear approximation ratio O(n/logn) was designed for the general problem in [BK01]. Thus, the improvement in approximation ratio for the dense instances given by this paper seems to be the largest known for any NP-hard constraint satisfaction problem. This paper extends the density sampler technique for graphs developed in [BFK00] to k-uniform hypergraphs for k ≥ 3, as the main tool to attack the dense MIN-Ek-LIN2 problems, or equivalently, k-ary versions of the Nearest Codeword problems, and the dense MIN-EkSat problems. The paper is organized as follows. In Section 2 we give the pre- liminaries and prove NP-hardness in exact setting of all the dense minimum satisfaction problems considered in this paper. Section 3 contains our main result on sampling k-uniform hypergraphs crucial for the rest of the paper. In Section 4, we design a PTAS for dense MIN-Ek-LIN2 and in Section 5 a PTAS for dense MIN-EkSAT for any k.
En savoir plus ### The 2013 Newton Institute Programme on polynomial optimization

– van Dam and Sotirov give some new bounding procedures for a rather gen- eral graph partitioning problem, and show that the resulting bounds dominate previous known ones, yet remain tractable in practice. – Pe˜ na et al. show that a wide family of PO problems can be reformulated as linear optimisation problems over completely positive cones. This extends a result of Burer, which applied only to certain mixed 0-1 quadratic programs. – de Klerk et al. analyse a known polynomial-time approximation scheme for ### Algebraic relaxations and hardness results in polynomial optimization and Lyapunov analysis

Sec. 5.2. Path-complete graphs and the joint spectral radius 117 theory. This is done in Section 5.2, where we define the notion of a path-complete graph (Definition 5.2) and prove that any such graph provides an approximation scheme for the JSR (Theorem 5.4). In Section 5.3, we give examples of families of path-complete graphs and show that many of the previously proposed techniques come from particular classes of simple path-complete graphs (e.g., Corollary 5.8, Corollary 5.9, and Remark 5.3.2). In Section 5.4, we characterize all the path- complete graphs with two nodes for the analysis of the JSR of two matrices. We determine how the approximations obtained from all of these graphs compare (Proposition 5.12). In Section 5.5, we study in more depth the approximation properties of a particular pair of “dual” path-complete graphs that seem to per- form very well in practice. Subsection 5.5.1 contains more general results about duality within path-complete graphs and its connection to transposition of ma- trices (Theorem 5.13). Subsection 5.5.2 gives an approximation guarantee for the graphs studied in Section 5.5 (Theorem 5.16), and Subsection 5.5.3 contains some numerical examples. In Section 5.6, we prove a converse theorem for the method of max-of-quadratics Lyapunov functions (Theorem 5.17) and an approx- imation guarantee for a new class of methods for proving stability of switched systems (Theorem 5.18). Finally, some concluding remarks and future directions are presented in Section 5.7.
En savoir plus ### Towards a general formal framework for polynomial approximation

2.2 Some other NPO problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Notations 4 4 The s ope of the paper and its main ontributions 5 II Roughing out a new approximation framework 7 5 Approximation hains: a generalization of the approximation algorithms 7 6 Approximation level 7 7 Convergen e and hardness threshold 8 8 Fun tional approximation-preserving redu tions 10 III A hieving approximation results in the new framework 11 9 Indu ed hereditary subgraph maximization problems 12 10 Maximum independent set 14 10.1 A rst improvement of the approximation of the maximum-weight independent set via theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
En savoir plus ### Polynomial approximation and graph-coloring

Based upon semi-lo al improvement, a renement of algorithm D_COLOR3 is devised in [24 ℄. The basi idea remains the same: one greedily nds a olle tion of independent sets of up to a ertain onstant size k + 1 , olors any of them with a new olor and removes all of them from the input graph; next she/he transforms the surviving graph into an instan e of k SC as we have seen in se tion 13 and approximately solves k SC in this instan e. In [24 ℄, k = 6 . The algorithm for 6SC proposed in ludes three phases. The rst one is totally greedy and onsists of nding a maximal olle tion of disjoint 6-sets in the initial 6SC-instan e (S, C) . The elements overed by this olle tion are next removed from C and the remaining sets are updated. Of ourse this update will eventually reate some 1-sets. The se ond phase is more restri tive than the rst one. Here a maximal olle tion of disjoint 5-sets, then 4-sets is onstru ted but with the restri tion that any su h set is hosen to make part of the olle tion only if its hoi e does not in rease the number of 1-sets reated during the rst phase. This is done greedily by onsidering a set and by examining if the removal of its elements will reate additional one sets. In what follows, we denote by STRICT_PHASE the pro edure implementing the se ond phase. The elements overed by olle tion so- onstru ted are removed from C and the remaining sets are updated. Finally, the third and last phase, applied in the surviving 3SC-instan e, is a semi-lo al (2,1)-improvement. In what follows, we denote by SL_OPT21 the algorithm repeatedly applying semi-lo al (2,1)- improvement until no su h improvement is possible.
En savoir plus ### ALGORITHMS FOR POSITIVE POLYNOMIAL APPROXIMATION

Problem 1.1. Find n + 1 interpolation points 0 ≤ x 0 < x 1 < · · · < x n ≤ 1 such that the polynomial interpolant p n ∈ P n defined by p n (x i ) = f (x i ) for all 0 ≤ i ≤ n satisfies p n ∈ P n + . Problem 1.1 has interest for pure numerical analysis purposes, and also because the question of having good characterization and convenient use of positive polyno- mials is central in applications and scientific computing. A non exhaustive list of references which reflects some of our own interests is: [1, 10] for cubic polynomials,  for automatization of the testing, [7, 5] with sum of squares characterization, [8, 9] on considerations on computer aided design with Bernstein and B´ezier curves, [15, 11, 14] for non negative numerical approximation in scientific computing for hyperbolic equa- tions and finally [5, 6] and therein for comprehensive references on polynomial theory. As Problem 1.1 is difficult to handle in full generality, it is convenient for theo-
En savoir plus ### Phase-field approximation for some branched transportation problems

is bounded independently of ε then the sequence of Radon measures µ ϕ ε weakly- ∗ converges up to subsequence to the measure H n−1 x {ϕ 6= 1}. As already stated in the introduction of this thesis, this fact is essential when approximating the Mumford - Shah functional as in the limit energy one aims at recovering the length of the jump set of some BV function which is contained in the set {ϕ 6= 1}. Later on some more general functionals have been introduced with different penalization of the jump set [BBB95] to model fractures. Recently [ABS99, DMOT16] other phase-field methods have dealt with the problem of efficiently approaching these energies. The main idea is that the limit ϕ rather then acquiring only the values {0, 1} should range over [0, +∞). In this chapter we follow this method to approximate any functional E h where h is a concave
En savoir plus ### Border Basis for Polynomial System Solving and Optimization

after a finite number of relaxations. The SDP solver used in this computation is mosek . The table records the number of variables (v), the number of inequality and equality constraints (c), the maximum degree of the constraints and of the poly- nomial to minimize (d), the number of minimizer points (sol), the maximal order (o), the maximal number of parameters (p), the maximal size of the moment ma- trices (s) in the SDP problems, and the total CPU time in seconds (t). ### Review and Analysis of some Metamodeling Techniques used in Optimization

In this paper we dis uss also the di ulties related to the onstru tion of these metamod- els. One of these di ulties is the lo ation of the observation points. Indeed, if the predi ted fun tion presents many variations, the observation points must be hosen su h that the in- formation obtained allows to reprodu e the behaviour of this fun tion. This means also a large number of the observation points. But the most important di ulty is the size of the data set. When this size is too large, it be omes too expensive to onstru t the metamodel. T o avoid this di ulty , some authors propose to onstru t a lo al metamodel, with only few observation points arround the desired one. The resulting metamodel is then heaper. But this requires enough available data in the vi inity of ea h predi ted point.
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