# Regularity, stability, and computation of equilibria

**FWF Project No. P26640-N25, (2014−2018), Project leader: V. Veliov**

Keywords: optimization, equilibria, numerical methods, optimal control, regularity, variational inequality

The notion of equilibrium plays a prominent role in science and engineering. Classically, equilibria are described by systems of equations, but if constraints are involved in the model, the equilibrium relations may take the form of a variational inequality (VI). Optimality conditions for optimization problems of different kinds, varying from mathematical programming to calculus of variations and optimal control, are standardly described by VIs. VIs may also arise in modeling of equilibrium processes in physics and economics. Examples are sweeping processes in continuum mechanics and Walrasian equilibria of product markets.

An important and desirable property of equilibria is their stability, that is, the property that an equilibrum does not disappear or change abruptly as a result of small changes in the model. This property is also often necessary for efficient computations of equilibria. The concept of metric regularity has emerged in the past 40 years (having its roots in classical works of Banach, Lyusternik, Graves, and others) as a powerful tool for investigation of equilibrium stability. This concept will be systematically employed in the project for VIs, parametric VIs, and differential VIs, describing equilibria in three different classes of models: static systems, exogenously changing systems, and endogenously evolving systems, respectively. Extended versions of a Walrasian model for economic equilibria will be used as workbench examples for the above three classes of VIs. The project consists of three main parts:

Metric regularity and conditioning, where the goal is to develop theoretical and numerical tools for estimation of the radius of metric regularity of classes of VIs, and for the workbench examples in particular. The radius of metric regularity gives information of how much a model can be disturbed before it experiences an abrupt change of its stability.

Parametric equilibria and path-following, where the goal is to develop predictor-corrector continuation methods for parametric VIs. These may ensure high-order approximations despite of the intrinsically non-smooth character of the underlying equilibrium problem. The direct application of predictor-corrector continuation methods will be compared, and possibly combined, with the semi-smooth (quasi-) Newton method applied to the reformulation of the VIs as non-smooth equations.

Differential variational inequalities and sweeping processes, where the development of high order numerical approximation schemes is the main goal. These will be implemented in new methods for solving optimal control problems, computing Nash equilibria in differential games, and computing dynamic economic equilibria.

The ultimate goal of the project is to achieve a better understanding of a broad class of VIs and stability of their equilibria, and to develop efficient numerical schemes for solving large-scale equilibrium problems. As a primal application we envisage economic equilibria, but applications to problems of optimal control or differential games that have a broader range of applications are also targeted.

**Publications based on the work on FWF Project P 26640-N25 till February, 2016**

[1] A.L. Dontchev, M.I. Krastanov, and V.M. Veliov. Omega-limit sets for differential inclusions. Analysis and Geometry in Control Theory and its Applications, P. Bettiol, P. Cannarsa, G. Colombo, M. Motta, F. Rampazzo (Eds.), Springer, 11, pp. 159--169, 2015. Also: Research Report 2014-09, ORCOS, TU Wien, 2014.

[2] S. Aseev and V.M. Veliov. Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions. Trudy Inst. Mat. i Mekh. UrO RAN, 20(3):41--57, 2014. and Proceedings of the Steklov Institute of Mathematics, 291(1)1:22–39, 2015. Also: Research Report 2014-06, ORCOS, TU Wien, 2014.

[3] A.L. Dontchev and V.M. Veliov. Regularity Properties of Mappings in Optimal Control. Control Processes., pp. 35--41, Proc. Internat. Conf. dedicated to the 90th anniversary of N.N. Krasovskii. Publisher: Institute of Mathematics and Mechanics, Ural Branch of the RAC, 2015. Also: Research Report 2015-06, ORCOS, TU Wien, 2015.

[4] A.O. Belyakov and V.M. Veliov. On Optimal Harvesting in Age-Structured Populations. (To appear.) Research Report 2015-08, ORCOS, TU Wien, 2015.

[5] V.M. Veliov. Relaxation of Euler-Type Discrete-Time Control System. In Large-Scale Scientific Computing, I. Lirkov, S. Margenov, J. Wasniewski (Eds.), Lecture Notes in Computer Science, 9374:134--141, Springer, 2015. Also: Research Report 2015-10, ORCOS, TU Wien, 2015.

[6] V.M. Veliov. Numerical Approximations in Optimal Control of a Class of Heterogeneous Systems. Computers and Mathematics with Applications, 70(11): 2652--2660, 2015. Also: Research Report 2015-01, ORCOS, TU Wien, 2015.

[7] R. Cibulka, A. L. Dontchev, J. Preininger, T. Roubal and V. Veliov. Kantorovich-type Theorems for Generalized Equations. (Submitted.) (see also Research Report 2015-16, ORCOS, TU Wien, 2015.)

[8] R. Cibulka, A.L. Dontchev and V.M. Veliov. Graves-type Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping. (Submitted) (see also Research Report 2015-14, ORCOS, TU Wien, 2015.)**Conference talks based on the work on FWF Project P 26640-N25 till February, 2016**

V.M. Veliov. On the optimal control on infinite horizon. Invited talk at conference "The 10th Conference on Dynamical Systems, Differential Equations and Applications", Madrid, Spain, July 07--11, 2014.

V.M. Veliov. Optimal control on infinite time-horizon: the maximum principle revisited. Invited plenary talk at conference "New Horizons in Optimal Control", Cascais, Portugal, Sept. 08--11, 2014.

V.M. Veliov. On a distributed optimal control problem of harvesting: periodicity analysis.}

Invited talk given at the "17th British-French-German Conference on Optimization", London, England, June 15--17, 2015.

V.M. Veliov. On the Relaxation of Discretized Differential Inclusions. Invited talk given at the "10th International Conference on Large Scale Scientific Computations", Sozopol, Bulgaria, June 08--12, 2015.

V.M. Veliov. On the regularity of two mappings in optimal control. Invited talk given at the "27th European Conference on Operational Research", Glasgow, July 12--15, 2015.

V.M. Veliov. Bi-metric Hölder regularity and stability of bang-bang optimal control. Talk given at the conference "Variational Analysis and Applications", Erice, Italy, August 29--Sept. 4, 2015.

V.M. Veliov. On the method of Kantorovich for non-smooth generalized equations. Invited plenary talk at conference "10th Annual Meeting of the Bulgarian Section of SIAM", Sofia, Bulgaria, December 21--22, 2015.

A. Dontchev. Newton-type methods in optimization and control. Optimization Seminar, ETH Zürich, Oct. 5, 2015.