Optimization and Equilibrium


Jorge Amaya, Luis BriceñoRafael CorreaFabián Flores, Pedro GajardoMarcos Goycoolea, Alejandro JofréHéctor RamírezDavid Salas, Emilio Vilches

Coordinators: Héctor Ramírez and Emilio Vilches

About the research group

Optimization and equilibrium models pervade mathematics and science. They have become central for decision making in industry. Our research has grown increasingly diverse as a result of our academic and industrial interactions.

Our competence in convex and nonlinear programming, nonsmooth dynamical systems, stochastic optimization, semi-algebraic optimization, variational analysis and game theory, combined with our engineers’ expertise, allow us to address relevant industrial problems from a multidisciplinary perspective.

Variational analysis and inequalities

Variational inequalities and differential inclusions appear consistently in mechanics, game theory and economics. Our research includes stability and approximation properties, as well as the design and evaluation of computationally efficient algorithms. This allows solving different kinds of equilibrium problems, especially in the sense of Nash and Walras, as well as other multi-objective optimization problems, where Pareto optimality plays a central role. On the other hand, by means of variational analysis techniques in a stochastic setting, we develop new models of financial equilibrium, for which we are able to provide existence results, as well as characterizations of equilibria via variational inequalities.

Conic and semi-infinite programming

We perform sensitivity analyses of conic and semi-infinite programming problems in order to assess their stability and robustness, which is vital for both theoretical and numerical reasons. The characterization of the (isolated) calmness of the solution map is of special importance, and can be expressed in terms of certain non-degeneracy conditions. We also found that the calmness of the solution mapping in linear semi-infinite optimization problems subject to canonical perturbations remains unaltered under Slater’s constraint qualification. A related line of research is the characterization of strong duality, and the evaluation of Karush-Kuhn-Tucker (KKT) optimality conditions, beyond constraint qualification. In particular, we study nonsmooth generalizations of Sard’s Theorem for Lipschitz-continuous vector-valued functions that can be expressed as selections of smooth functions, and its consequences in the genericity of the KKT conditions in the semi-infinite optimization problem.

Dynamical systems and algorithms of gradient type

Continuous-time dynamical systems are an important source of insight into the development of fast and proficient algorithms to solve optimization problems, and are interesting in their own right. Arc-length reparameterization of orbits of possibly nonsmooth dynamics reveal properties of the intrinsic geometry of the level sets. We have found that this prevents the orbits from stopping at inessential singularities, and continue their search for global optimizers. We are especially interested in the close connections with geometric properties of the functions, which impact the convergence rates of algorithms. We also study second-order systems, with emphasis on the convex and quasiconvex case. On the one hand, a comparison between first and second order solutions, allows detecting hidden Lyapunov functions, which encode asymptotic properties of the orbits. On the other hand, we have found that inertial dynamics are at the core of accelerated algorithms, and that the study of the continuous-time models are useful in their analysis. Finally, we study existence of solutions, sensitivity and asymptotic behavior of the orbits of tame sweeping processes.

Industrial applications

The principles and methods of mathematical optimization are applied in order to generate and implement solutions to industrial problems of high impact. We characterize the optimal feeding strategy for a bioreactor to clean up a lake in minimal time, especially when the pollutant is not homogeneously distributed. We also analyze fisheries to maximize the harvesting, subject to sustainability criteria. We develop mathematical models for the optimal location of schools and other facilities, under stochastic evolution of the demand. For the urban train system, we design policies to improve the efficiency in the use of human resources and energy. We analyze block models to determine the optimal sequence of ore extraction in underground and open-pit mines. Finally, we model complex production and distribution networks for the energy market.