Lecturer
Boris Mordukhovich, Wayne Sate University, Detroit, Michigan, USA
Schedule and place
This course will take place in Building C - Stadscampus, University of Antwerp, Prinsstraat 13, 2000 Antwerpen
Schedule: 08, 09, 10, 11, and 12 April 2024 (Monday to Friday), 09:00-11:30, including a coffee break.
The course is organized in the context of a one-week workshop on Nonsmooth Optimization and Applications. SOCN students are welcome to attend the rest of the workshop, but then a distinct registration is required.
Description
This course consists of five lectures devoted to advanced tools of variational analysis and generalized differentiation with their applications to qualitative and algorithmic aspects of optimization theory and optimal control.
LECTURE 1 (April 8) VARIATIONAL ANALYSIS: AN OVERVIEW. This lecture contains a brief overview of basic principles and constructions of variational analysis and generalized differentiation with reviewing some underlying results and areas of applications. In particular, we axiomatically define subdifferentials of nonsmooth functions and normal to closed sets with their major realizations, present the extremal principle and its applications to generalized Lagrange multiplier rules, and formulate necessary optimality conditions for general problems of optimal control.
LECTURE 2 (April 9) CRITICALITY OF LAGRANGE MULTIPLIERS IN CONSTRAINED OPTIMIZATION WITH APPLICATIONS TO SQP. This lecture is devoted to a new theory of critical and noncritical Lagrange multipliers for general problems of constrained optimization, including conic programming. Using advanced machinery of variational analysis a generalized differentiation, we present second-order chacterizations of critical and noncritical multiplies with their applications to the sequential quadratic programming method (SQP) in problems of constrained optimization.
LECTURE 3 (April 10) INEXACT REDUCED GRADIENT AND PROXIMAL METHODS IN NONCONVEX OPTIMIZATION. In this lecture, we first discuss new linesearch methods with inexact gradient information for finding stationary points of nonconvex smooth functions. Some abstract convergence results for a broad class of linesearch methods are established. A general scheme for inexact reduced gradient (IRG) methods is presented, where the errors in the gradient approximation automatically adapt with the magnitudes of the exact gradients. The sequences of iterations are shown to obtain stationary accumulation points when different stepsize selections are employed. Convergence results with constructive convergence rates for the developed IRG methods are established under the Kurdyka- Lojasiewicz property. The obtained results for the IRG methods are confirmed by encouraging numerical experiments, which demonstrate advantages of automatically controlled errors in IRG methods over other frequently used error selections. Then we develop inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term.
LECTURE 4 (April 11) GENERALIZED NEWTONIAN ALGORITHMS IN NONSMOOTH OPTIMIZATION. This lecture is devoted to applications of second-order variational analysis and generalized differentiation to the design and justification of novel generalized Newtonian algorithms. First we present the results on locally superlinearly convergent generalized Newton methos and then proceed with globally convergent coderivative-based algorithms of the damped Newton method and of the regularized Newton method designed via the generalized Hessian. Efficient conditions for global superlinear convergence of these algorithms are obtained for problems of convex composite optimization with applications to Lasso and related problems of machine learning.
LECTURE 5 (April 12) OPTIMAL CONTROL OF SWEEPING PROCESSES WITH APPLICATIONS TO ROBOTICS AND TRAFFIC EQUILIBRIA. The final lecture of this course is devoted to applications of variational analysis to a new and challenging class of optimal control problems for the so-called sweeping (Moreau) processes, which are governed by discontinuous differential inclusions and dynamic quasi-variational inequalities. We develop the method of finite-difference/discrete approximations to the study of such problems and then derive in this way necessary optimality conditions for discrete-time and continuous-time systems with applications to some models of robotics, traffic equilibria, etc
Course material
TBD
Evaluation
TBD