Title: The turnpike property: a classical feature of optimal control problems revisited

Speaker: Lars Grüne (University of Bayreuth)

Date and Time: Wed May 04 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

The turnpike property describes a particular behavior of optimal control problems that was first observed by Ramsey in the 1920s and by von Neumann in the 1930s. Since then it has found widespread attention in mathematical economics and control theory alike. In recent years it received renewed interest, on the one hand in optimization with partial differential equations and on the other hand in model predictive control (MPC), one of the most popular optimization based control schemes in practice. In this talk we will first give a general introduction to and a brief history of the turnpike property, before we look at it from a systems and control theoretic point of view. Particularly, we will clarify its relation to dissipativity, detectability, and sensitivity properties of optimal control problems in both finite and infinite dimensions. In the final part of the talk we will explain why the turnpike property is important for analyzing the performance of MPC.

Title: Approximating convex bodies using multiple objective optimization

Speaker: Andreas Löhne (Friedrich Schiller University Jena)

Date and Time: Wed Apr 27 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

The problem to compute a polyhedral outer and inner approximation of a convex body can be reformulated as a problem to solve approximately a convex multiple objective optimization problem. This extends a previous result showing that multiple objective linear programming is equivalent to compute a $V$-representation of the projection of an $H$-polyhedron. These results are also discussed with respect to duality, solution methods and error bounds.

Title: Extensions of Constant Rank Qualification Constrains condition to Nonlinear Conic Programming

Speaker: Héctor Ramírez (Universidad de Chile)

Date and Time: Wed Apr 13 2022, 11:00 AEST (Register here for remote connection via Zoom)

Abstract:

We present new constraint qualification conditions for nonlinear conic programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithms includes, for instance, general forms of augmented Lagrangian, sequential quadratic programming, and interior point methods. We also compare these new conditions with some of the existing ones, including the nondegeneracy condition, Robinson’s constraint qualification, and the metric subregularity constraint qualification. Finally, we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of the latter is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson’s constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson’s condition.

Title: Extending the proximal point algorithm beyond convexity

Speaker: Sorin-Mihai Grad (ENSTA Paris)

Date and Time: Wed Apr 06 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

Introduced in in the 1970’s by Martinet for minimizing convex functions and extended shortly afterwards by Rockafellar towards monotone inclusion problems, the proximal point algorithm turned out to be a viable computational method for solving various classes of (structured) optimization problems even beyond the convex framework. In this talk we discuss some extensions of proximal point type algorithms beyond convexity. First we propose a relaxed-inertial proximal point type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions whose variables lie in finitely dimensional linear subspaces, that can be extended to equilibrium functions involving such functions. Then we briefly discuss another generalized convexity notion for functions we called prox-convexity for which the proximity operator is single-valued and firmly nonexpansive, and see that the standard proximal point algorithm and Malitsky’s Golden Ratio Algorithm (originally proposed for solving convex mixed variational inequalities) remain convergent when the involved functions are taken prox-convex, too. The talk contains joint work with Felipe Lara and Raúl Tintaya Marcavillaca (both from University of Tarapacá).