Nash Equilibrium Problems of Polynomials

This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Lasserre type Moment-SOS relaxations are used to solve them. Under generic assumptions, the method can find a Nash equilibrium if there is one. Moreover, it can find all Nash equilibria if there are finitely many ones of them. The method can also detect nonexistence if there is no Nash equilibrium.Comment: 26 page

Positivity of the universal pairing in 3 dimensions

Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2+1 dimensional TQFTs. The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) <= max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B. The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.Comment: 83 pages, 21 figures; version 3: incorporates referee's comments and correction

Macroeconomic impact of the risk-taking channel : evidence from SVAR with nonnormal residuals

The identifying restrictions of an earlier VAR model are validated to assess the macroeconomic impact of the risk-taking channel of monetary policy in the U.S. Structural shocks are obtained by exploiting the nonnormality of residuals. The data is found to object to the previously imposed recursive ordering, but a different recursive ordering is supported. Based on the resulting impulse responses, there is no strong and significant evidence of the risk-taking channel during the sample period. This finding is in contrast with both the predictions of the underlying theoretical model and previous empirical findings

Numerical algebraic geometry approach to polynomial optimization, The

2017 Summer.Includes bibliographical references.Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science and engineering. This research focused on finding global solutions to constrained polynomial optimization problems of moderate size using NAG methods. The benefit of employing a NAG approach to nonlinear optimization problems is that every critical point of the objective function is obtained with probability-one. The NAG approach to global optimization aims to reduce computational complexity during path tracking by exploiting structure that arises from the corresponding polynomial systems. This thesis will consider applications to systems biology and life sciences where polynomials solve problems in model compatibility, model selection, and parameter estimation. Furthermore, these techniques produce mathematical models of large data sets on non-euclidean manifolds such as a disjoint union of Grassmannians. These methods will also play a role in analyzing the performance of existing local methods for solving polynomial optimization problems

Decentralized blocking zeros in the control of large scale systems

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Science of Bilkent Univ., 1992.Thesis (Ph. D.) -- Bilkent University, 1992.Includes bibliographical references.lu lliis lliesi.s, a luuiiber ot syiithe.sis problems i'or linear. ninc-invariauL, iiiiite-cliuieiiSioiial sysiems are adclres.se(l. It i.s sliown that tlie lu'w concejU of (l·.': m inili zed blocking zeros \s as fmidaineiital to controller .synthesis problems for large scale systems as the concept of decentralized fixed modes. The main problems considered are (i) decentralized stabilization problem, (ii) decentralized strong stabilization problem, and (iii) decentralized concurrent stabilization problem. 7'he dtcenIralized siabUizaiion problem is a fairly well-understood controller synthesis problem for which many synthesis methods exist. Here, we give a new .synthesis procedure via a proper stable fractional approach and focus our attention on the generic solvability and characitnzalion of all solutions. The decenlralized strong .stabilization problem is the problem of stabilizing a .systeni using stable local controllers. In this problem, the .set of decentralized blocking zeros play an essential role and it turns out that the problem has a solution in case tlie poles and the real nonnegative decentralized blocking zeros have parity interlacing property. In the more general problem of decentralized stabilization problem with minimum number of unstable controller poles, it is shown tliat this minimum number is determined by the nuiid.H-»r of odd distributions of plant poles among the real nonnegative decentralized blocking zeros. The decentralized concurrent stabilization problem is a special type of simultaneous stabilization problem using a decentralized controller. Tliis problem is of interest, since many large scale synthesis problems turn out to be its special cases. A complete solution to decentralized concurrent stabilization problem is obtained, where again the decentralized blocking zeros play a central role. Three problems that have receiviHİ wide atteiuion in tlie literature of large scale .systems: stabilization o f composite systems using locally :>tabilizing subsystem controllers, stabilization uf composite system.^ na the slabilization o f mam diagonal transfer matrices, and rcliablt decentralized siabilizaiion problem are solved by a specialization of oiir main result on decentralized concurrent stabilization problem.Ünyelioğlu, Konur AlpPh.D