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Pavlo O. Kasyanov

Publications and source records attributed to Pavlo O. Kasyanov.

30 records · Page 2Linked to original sources
arXiv2017

Continuity of Equilibria for Two-Person Zero-Sum Games with Noncompact Action Sets and Unbounded Payoffs

This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.

math.OC↗
arXiv2015

Uniform Fatou's Lemma

Fatou's lemma is a classic fact in real analysis that states that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou's lemma, and it holds under the minor assumption that all the integrals are well-defined. The uniform Fatou's lemma improves the classic Fatou's lemma in the following directions: the uniform Fatou's lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou's lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou's lemma may indeed provide a more accurate inequality than the classic Fatou's lemma; (b) the uniform Fatou's lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence.

math.FA↗
arXiv2014

Continuity of Minima: Local Results

This paper compares and generalizes Berge's maximum theorem for noncompact image sets established in Feinberg, Kasyanov and Voorneveld (2014) and the local maximum theorem established in Bonnans and Shapiro (2000).

math.OC↗
arXiv2014

Convergence of Probability Measures and Markov Decision Models with Incomplete Information

This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise converges, and convergence in the total variation. First, it describes and compares necessary and sufficient conditions for these types of convergence, some of which are well-known, in terms of convergence of probabilities of open and closed sets and, for the probabilities on the real line, in terms of convergence of distribution functions. Second, it provides % convenient criteria for weak and setwise convergence of probability measures and continuity of stochastic kernels in terms of convergence of probabilities defined on the base of the topology generated by the metric. Third, it provides applications to control of Partially Observable Markov Decision Processes and, in particular, to Markov Decision Models with incomplete information.

math.PR↗
arXiv2014

Partially Observable Total-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities

This paper describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs) with Borel state, observation, and action sets and with the expected total costs. Action sets may not be compact and one-step cost functions may be unbounded. The introduced conditions are also sufficient for the validity of optimality equations, semi-continuity of value functions, and convergence of value iterations to optimal values. Since POMDPs can be reduced to Completely Observable Markov Decision Processes (COMDPs), whose states are posterior state distributions, this paper focuses on the validity of the above mentioned optimality properties for COMDPs. The central question is whether transition probabilities for a COMDP are weakly continuous. We introduce sufficient conditions for this and show that the transition probabilities for a COMDP are weakly continuous, if transition probabilities of the underlying Markov Decision Process are weakly continuous and observation probabilities for the POMDP are continuous in the total variation. Moreover, the continuity in the total variation of the observation probabilities cannot be weakened to setwise continuity. The results are illustrated with counterexamples and examples.

math.OC↗
arXiv2013

Fatou's Lemma for Weakly Converging Probabilities

Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.

math.PR↗
arXiv2013

Berge's Maximum Theorem for Noncompact Image Sets

This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the results from E.A. Feinberg, P.O. Kasyanov, N.V. Zadoianchuk, "Berge's theorem for noncompact image sets," J. Math. Anal. Appl. 397(1)(2013), pp. 255-259 on the extension to noncompact image sets of another Berge's theorem, that states semi-continuity of value functions. Here we explain that the notion of a $\K$-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a $\K\N$-inf-compact function ($\N$ stands for "nets" in $\K$-inf-compactness), which coincides with $\K$-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.

math.GN↗
arXiv2012

Structure and regularity of the global attractor of a reaction-di§usion equation with non-smooth nonlinear term

In this paper we study the structure of the global attractor for a reaction- di§usion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.

math.DS↗
arXiv2012

Berge's Theorem for Noncompact Image Sets

For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this function taken over the image sets. It assumes that the image sets are compact. For Hausdorff topological spaces, this paper extends Berge's theorem to set-valued mappings with possible noncompact image sets and studies relevant properties of minima.

math.GN↗
arXiv2012

Average-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities

This paper presents sufficient conditions for the existence of stationary optimal policies for average-cost Markov Decision Processes with Borel state and action sets and with weakly continuous transition probabilities. The one-step cost functions may be unbounded, and action sets may be noncompact. The main contributions of this paper are: (i) general sufficient conditions for the existence of stationary discount-optimal and average-cost optimal policies and descriptions of properties of value functions and sets of optimal actions, (ii) a sufficient condition for the average-cost optimality of a stationary policy in the form of optimality inequalities, and (iii) approximations of average-cost optimal actions by discount-optimal actions.

math.OC↗