Title: Constrained Structured Optimization and Augmented Lagrangian Proximal Methods

Speaker: Alberto De Marchi (Universität der Bundeswehr München)

Date and Time: Wed May 25 2022, 17:00 AEST (Register here for remote connection via Zoom)

Abstract:

In this talk we discuss finite-dimensional constrained structured optimization problems and explore methods for their numerical solution. Featuring a composite objective function and set-membership constraints, this problem class offers a modeling framework for a variety of applications. A general and flexible algorithm is proposed that interlaces proximal methods and safeguarded augmented Lagrangian schemes. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. Adopting a proximal gradient method with an oracle as a formal tool, it is demonstrated how the inner subproblems can be solved by off-the-shelf methods for composite optimization, without introducing slack variables and despite the appearance of set-valued projections. Illustrative examples show the versatility of constrained structured programs as a modeling tool and highlight benefits of the implicit approach developed. A preprint paper is available at arXiv:2203.05276.

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.