Optimization and Equilibrium


CMM: Jorge AmayaRafael Correa, Aris Daniilidis, Juan EscobarAbderrahim HantouteAlejandro Jofré, Héctor Ramírez

UTFSM: Luis Briceño, Pedro Gajardo, Juan PeypouquetAdriana Piazza

U. Concepción: Fabián Flores

UAI: Bernardo Pagnoncelli

About the research group

Optimization and equilibrium models pervade mathematics and science. They have also become central for decision making in industry. Our research has grown steadily with an increasing diversification derived from academic and industrial interactions.

Our basic competences in mathematical programming, convex analysis, calculus of variations, variational inequalities, and game theory, combined to the expertise of engineers, allow us to undertake multidisciplinary work such as short-term planning of a mine, mechanism design in the electricity generation market, optimization of bioprocesses involved in wastewater treatment, sustainable exploitation of fisheries, or competitive equilibrium in telecommunications.

The main research areas considered by this group are: convex and nonlinear programming, nonsmooth dynamical systems, stochastic optimization, semialgebraic optimization, variational analysis, game theory, Pareto optimization, management of natural resources, optimal control, and energy system,

Variational analysis and Inequalities, game theory and its applications to economics.

We have developed a series of applications of lopsided convergence to equilibrium problems (Nash, Walras). Variational Inequalities and inclusions, including stability and approximation results, all based on a max-min representation of these problems and the lopsided convergence notions were analyzed in a previous paper with R. Wets in Mathematical Programming. On the other hand, by using variational analysis techniques in a stochastic setting, we developed a new model on financial equilibrium. For this model, we have proved existence of equilibrium points and we provided a characterization of equilibria via variational inequalities. At the same time we have investigated questions related to the existence of Pareto optimal points as well as free disposal conditions for nonconvex economies on infinite-dimensional commodity spaces.

Mathematical programming, semi-infinite optimization and applications.

Concerning sensitivity analysis of cone programming, we have studied the behavior of solutions to perturbed generalized equations (in short, GE) over conic constraints. More specifically, we have fully characterized the isolated calmness of the solution map of this GE provided the nominal solution is nondegenerate and the conic constraints satisfy an extended polyhedricity property around this nominal solution. We are currently investigating extensions of these results in two directions: applying the same methodology to study other related sensitivity properties (such as full stability) and dropping the assumption of nondegeneracy. The latter implies dealing with possibly multiple Lagrange multipliers, which could highly increases the complexity of our analysis. In a parallel line of research, we have established a «preparatory» Sard-type theorem for strongly critical values of smooth functions with a partial affine structure, and we have used it to obtain a nonsmooth generalization of the Sard theorem for (vector- valued) Lipschitz functions that can be expressed as finite selections of smooth functions (more generally, continuous selections over a compact countable set). This powerful result recovers readily the classical Sard theorem and is paramount for establishing the genericity of the Karush-Kuhn- Tucker conditions in the semi-infinite optimization problem. We have also obtained several characterizations for the validity of strong duality property; and the fulfillment of KKT optimality conditions, beyond constraints qualification, in differentiable optimization.

Subdifferential calculus and stability of optimization problems.

We established new subdifferential calculus rules for the sum of convex functions defined on normed spaces. This is achieved by means of a condition relying on continuity behavior of the inf-convolution of the corresponding conjugate functions, with respect to any topology lying between the norm and the weak* topologies on the dual space. Such a condition turns out to be also necessary in Banach spaces. These results extend classical results, which have been previously obtained by Hiriart-Urruty & Phelps and by Thibault. Concerning stability aspects of optimization problems, we characterize the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this work, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.

Gradient dynamical systems and nonsmooth dynamics.

Parametrizing by arc-length the orbits of a gradient dynamical system of a smooth potential f we obtain a system which is much more intrinsic to the geometry of the level sets: the orbits of the system may reach a singularity in finite time and continue from there onward while not stopping at inessential singularities. A particularly important situation arises when the function f is quasiconvex, where we may even drop the smoothness assumption on f and instead seek, absolutely continuous curves x(t) which satisfy a differential inclusion based on the normal cones to the sublevel sets. We established that this system (under very mild assumptions on the potential) always admits Lipschitz continuous trajectories starting from any point. Moreover maximal trajectories are either unbounded or converge to the global minimum of the function. As a step forward, we study the classical sweeping process (introduced by J.-J. Moreau): existence of solutions, sensitivity and asymptotic behavior of the orbits in the tame case.

Self-contracted curves.

We have pursued our investigations to the second order study of dynamical systems of gradient type, with emphasis on the convex and quasiconvex case. A comparison between first and second order solutions, allows one to detect hidden Lyapunov functions, which are helpful for the study of the asymptotic behavior of the orbits and eventually come up with new properties of the convex systems. This topic obviously relates to the general study of convex foliations treating the particular case of smooth representations. It also underlines the paramount notion of self-contracted curves. This latter notion is of crucial importance in convex optimization: among its numerous consequences, it relates to estimates for the convergence of the proximal algorithm and has been the object of a thorough study since the last two years.

Applications in bioprocess.

We have also studied several problems involving bioprocess and the pollution of water reservoirs, such as lakes. For instance, we have fully characterized the optimal (regarding a minimal time criterium) feeding strategy for a bioreactor to clean up a lake provided that the pollutant is homogeneously distributed in the lake. The latter condition being too restrictive for real-life applications, we are currently studying this problem for nonhomogeneous models obtained via the so-called “compartment models” approach. Solving this problem and using our previous (homogeneous) solution for improving simulations of specific PDE models where the geometry of the lake plays a crucial role is challenging. We can typically observe that the solution obtained in the homogeneous case is in a way better than other “naive” solutions for these two nonhomogeneous type models. However, to compute, either analytically or numerically the optimal solution of the proposed nonhomogeneous models seems a difficult and challenging task.