The 0-1 inverse maximum stable set problem

Given an instance of a weighted combinatorial optimization problem and its feasible solution, the usual inverse problem is to modify as little as possible (with respect to a fixed norm) the given weight system to make the giiven feasible solution optimal. We focus on its 0-1 version, which is to modify as little as possible the structure of the given instance so that the fixed solution becomes optimal in the new instance. In this paper, we consider the 0-1 inverse maximum stable set problem against a specific (optimal or not) algorithm, which is to delete as few vertices as possible so that the fixed stable set S* can be returned as a solution by the given algorithm in the new instance. Firstly, we study the hardness and approximation results of the 0-1 inverse maximum stable set problem against the algorithms. Greedy and 2-opt. Secondly, we identify classes of graphs for which the 0-1 inverse maximum stable set problem can be polynomially solvable. We prove the tractability of the problem for several classes of perfect graphs such as comparability graphs and chordal graphs.Combinatorial inverse optimization, maximum stable set problem, NP-hardness, performance ratio, perfect graphs.

Maximum Entropy Kernels for System Identification

A new nonparametric approach for system identification has been recently proposed where the impulse response is modeled as the realization of a zero-mean Gaussian process whose covariance (kernel) has to be estimated from data. In this scheme, quality of the estimates crucially depends on the parametrization of the covariance of the Gaussian process. A family of kernels that have been shown to be particularly effective in the system identification framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy properties of a related family of kernels, the Tuned/Correlated (TC) kernels, have been recently pointed out in the literature. In this paper we show that maximum entropy properties indeed extend to the whole family of DC kernels. The maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the DC kernel. In particular, we prove that the DC kernel admits a closed-form factorization, inverse and determinant. These results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the DC estimator.Comment: Extends results of 2014 IEEE MSC Conference Proceedings (arXiv:1406.5706

A mean field method with correlations determined by linear response

We introduce a new mean-field approximation based on the reconciliation of maximum entropy and linear response for correlations in the cluster variation method. Within a general formalism that includes previous mean-field methods, we derive formulas improving upon, e.g., the Bethe approximation and the Sessak-Monasson result at high temperature. Applying the method to direct and inverse Ising problems, we find improvements over standard implementations.Comment: 15 pages, 8 figures, 9 appendices, significant expansion on versions v1 and v

Dual and inverse formulations of constrained extremum problems

AbstractIn this paper we consider constrained extremum problems of th form Pp:infu∈t−1(p)f(u) where f and t are continuously differentiable functionals on a reflexive Banach space V and where t-1(p) denotes the level set of the functional t with value p ϵ R.Related to problems Pp we investigate inverse extremum problems, which are extremum problems for the functional t on level sets of the functional f. Under conditions that guarantee the existence of solutions of Pp, let h(p) denote the value of f at such a solution. If h is a (locally) convex function at some p̄ ϵ R, we show that it is possible to define a dual problem of Pp̄. This dual problem is a saddle-point formulation for the functional V × RЭ(u,μ)→f(u)−μ[t(u)−p̄]: for some extreme value μ̄ (which is the Lagrange multiplier of a solution of Pp̄) the solutions of Pp̄ are precisely the (local) minimal points of the functional f − μ̄t on V.It is shown how these results can be used to describe solution branches of nonlinear eigenvalue problems (of semilinear elliptic type) with a global parameter, such as p ϵ R, instead of with the eigenvalue as a parameter

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for $3\times 3$ matrices.Comment: 27 page