Proceedings of the Steklov Institute of Mathematics最新文献

The paper considers the convex hull of a set of schedules for servicing identical requests by parallel devices. Precedence conditions are given on the set of requests. All requests enter the service queue simultaneously and have the same service duration. Interruptions in request servicing are prohibited. Time is discrete. The polyhedral properties of some previously constructed classes of valid inequalities are studied. The “depth” cuts are compared, and the strongest subclasses of cuts are found. The relative position of the schedule polytope and hyperplanes generated by inequalities is also studied.

Recently, O. Svensson and V. Traub have provided the first proof of the polynomial-time approximability of the asymmetric traveling salesman problem (ATSP) in the class of constant-factor approximation algorithms. Just as the famous Christofides–Serdyukov algorithm for the symmetric routing problems, these breakthrough results, applied as a “black box,” have opened an opportunity for developing the first constant-factor polynomial-time approximation algorithms for several related combinatorial problems. At the same time, problems have been revealed in which this simple approach, based on reducing a given instance to one or more auxiliary ATSP instances, does not succeed. In the present paper, we extend the Svensson–Traub approach to the wider class of problems related to finding a minimum-weight cycle cover of an edge-weighted directed graph with an additional constraint on the number of cycles. In particular, it is shown for the first time that the minimum weight cycle cover problem with at most (k) cycles admits polynomial-time approximation with constant factor (max{22+varepsilon,4+k}) for arbitrary (varepsilon>0).

The paper presents an overview of methods for solving an ill-posed problem of constrained convex quadratic minimization based on the Fejér-type iterative methods, which widely use the ideas and approaches developed in the works of I. I. Eremin, the founder of the Ural research school of mathematical programming. Along with a problem statement of general form, we consider variants of the original problem with constraints in the form of systems of equalities and inequalities, which have numerous applications. In addition, particular formulations of the problem are investigated, including the problem of finding a metric projection and solving a linear program, which are of independent interest. A distinctive feature of these methods is that not only convergence but also stability with respect to errors in the input data are established for them; i.e., the methods generate regularizing algorithms in contrast to the direct methods, which do not have this property.

In this paper, we investigate a problem of optimal control over a finite time interval for a linear systemwith constant coefficients and a small parameter in the initial data in the class of piecewise continuous controlswith smooth geometric constraints. We consider a terminal convex performance index. We substantiate the limit relationsas the small parameter tends to zero for the optimal value of the performance index and for the vector generatingthe optimal control in the problem. We show that the asymptotics of the solution can be of complicated nature. Inparticular, it may have no expansion in the Poincaré sense in any asymptotic sequence of rational functions of thesmall parameter or its logarithms.

We develop a mathematical technique of Young dual variational inequalities, which are used to model market equilibrium in a network of production clusters that are heterogeneous from a technological point of view. Two formulations of the problem are considered: for a closed system with a given constraint on resources and for an open system in which resources can be supplied from outside at given prices. A theorem is proved on the existence of a solution to the variational inequality corresponding to market equilibrium in an open system.

The paper is concerned with sharp Carlson type inequalities of the form(|w(cdot)x(cdot)|_{L_{q}(T)}leq K|w_{0}(cdot)x(cdot)|_{L_{p}(T)}^{ gamma}max_{1leq jleq n}|w_{j}(cdot)x(cdot)|_{L_{r}(T)}^{1-gamma},)where (T) is a cone in (mathbb{R}^{d}) and the weight functions (w_{j}(cdot)), (j=1,mathinner{ldotpldotpldotp},n), are homogeneous with some symmetry property.

Earlier, the author described up to conjugacy all pairs ((A,B)) of nilpotent subgroups of a finite group (G) with socle (L_{2}(q)) for which (Acap B^{g}neq 1) for any element of (G). A similar description was obtained by the author later for primary subgroups (A) and (B) of a finite group (G) with socle (L_{n}(2^{m})). In this paper, we describe up to conjugacy all pairs ((A,B)) of nilpotent subgroups of a finite group (G) with simple socle from the “Atlas of Finite Groups” for which (Acap B^{g}neq 1) for any element (g) of (G). The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group (G) for any nilpotent subgroups (N) contains an element (g) such that (Ncap N^{g}=1).

We consider a decomposition approach to the problem of finding the orthogonal projection of a given point onto a convex polyhedral cone represented by a finite set of its generators. The reducibility of an arbitrary linear optimization problem to such projection problem potentially makes this approach one of the possible new ways to solve large-scale linear programming problems. Such an approach can be based on the idea of a recurrent dichotomy that splits the original large-scale problem into a binary tree of conical projections corresponding to a successive decomposition of the initial cone into the sum of lesser subcones. The key operation of this approach consists in solving the problem of projecting a certain point onto a cone represented as the sum of two subcones with the smallest possible modification of these subcones and their arbitrary selection. Three iterative algorithms implementing this basic operation are proposed, their convergence is proved, and numerical experiments demonstrating both the computational efficiency of the algorithms and certain challenges in their application are performed.

We consider an inclusion (widetilde{y}in F(x)) with a multivalued mapping acting in spaces with vector-valued metricswhose values are elements of cones in Banach spaces and can be infinite. A statement about the existence of a solution (xin X)and an estimate of its deviation from a given element (x_{0}in X) in a vector-valued metric are obtained. This result extendsthe known theorems on similar operator equations and inclusions in metric spaces and in the spaces with (n)-dimensional metricto a more general case and, applied to particular classes of functional equations and inclusions, allows to get less restrictive,compared to known, solvability conditions as well as more precise estimates of solutions. We apply this result to the integral inclusion(widetilde{y}(t)in f(t,intop_{a}^{b}varkappa(t,s)x(s),ds,x(t)), tin[a, b],)where the function (widetilde{y}) is measurable, the mapping (f) satisfies the Carathéodory conditions, and the solution (x) isrequired to be only measurable (the integrability of (x) is not assumed).

Let (mathfrak{F}) be a formation, and let (G) be a finite group. A subgroup (H) of (G) is called (mathrm{K}mathfrak{F})-subnormal (submodular) in (G) if there is a subgroup chain (H=H_{0}leq H_{1}leqmathinner{ldotpldotpldotp}leq H_{n-1}leq H_{n}=G) such that, for every (i) either (H_{i}) is normal in (H_{i+1}) or (H_{i+1}^{mathfrak{F}}leq H_{i}) ((H_{i}) is a modular subgroup of (H_{i+1}), respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is (mathrm{K}mathfrak{U}_{1})-subnormal. Here (mathfrak{U}_{1}) is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation (mathfrak{F}), every solvable (mathrm{K}mathfrak{F})-subnormal subgroup of a group (G) is contained in the solvable radical of (G). We also obtain a series of applications of these results to the investigation of groups factorized by (mathrm{K}mathfrak{F})-subnormal and submodular subgroups.