Mathematics of Computation and Optimisation

Title: Algebraic Perspectives on Signomial Optimization

Speaker: Mareike Dressler (University of New South Wales)

Date and Time: Wed May 11 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of “degree” that is central to polynomial optimization theory. In this talk, I introduce the concept of signomial rings that allows to reclaim that principle and explain how this leads to complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. In the first part of the talk, I discuss the Positivstellensatz underlying the lower bounds. It relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity of the feasible set or Archimedeanity of its representing signomial inequalities. Numerical examples are provided to illustrate the performance of the hierarchy on problems in chemical engineering and reaction networks. In the second part, I provide a language for and basic results in signomial moment theory that are analogous to those in the rich moment-SOS literature for polynomial optimization. That theory is used to turn (hierarchical) inner-approximations of signomial nonnegativity cones into (hierarchical) outer-approximations of the same, which eventually yields the upper bounds for signomial optimization. This talk is based on joint work with Riley Murray.

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).

Title: Regularized dynamical systems associated with structured monotone inclusions

Speaker: Pham Ky Anh (Vietnam National University)

Date and Time: Wed Mar 30 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

In this report, we consider two dynamical systems associated with additively structured monotone inclusions involving a multi-valued maximally monotone operator A and a single-valued operator B in real Hilbert spaces. We established strong convergence of the regularized forward-backward and regularized forward – backward–forward dynamics to an “optimal” solution of the original inclusion under a weak assumption on the single-valued operator B. Convergence estimates are obtained if the composite operator A + B is maximally monotone and strongly (pseudo)monotone. Time-discretization of the corresponding continuous dynamics provides an iterative regularization forward-backward method or an iterative regularization forward-backward-forward method with relaxation parameters. Some simple numerical examples were given to illustrate the agreement between analytical and numerical results as well as the performance of the proposed algorithms.

Title: Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors

Speaker: Shawn Wang (The University of British Columbia)

Date and Time: Wed Mar 23 2022, 11:00 AEST (Register here for remote connection via Zoom)

Abstract:

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. We extend Simons’s lemma to closed convex functions, show its connections to Attouch-Théra duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings. Joint work with H. Bauschke