Session: Nonsmooth Optimization - Theory and Algorithms
Chair: Shaoning Han
Cluster:
Talk 1: Analysis of a Class of Heaviside Composite Minimization Problems
Speaker: Shaoning Han
Abstract: The minimization of nonclosed functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. In this talk, we first present difficulties of tackling such problems using existing optimization techniques. Then we establish stationarity conditions for the problem, and introduce a sufficient condition, termed local convex-like property, under which the proposed stationary point is a local minimizer. Finally, we briefly discuss solution strategies via lifting and reformulation techniques. For details of this work, please refer to "Han S, Cui Y, Pang J-S (2024) Analysis of a class of minimization problems lacking lower semicontinuity. Mathematics of Operations Research."
Talk 2: Non-smooth stochastic gradient descent using smoothing functions
Speaker: Jingfu Tan
Abstract: This work is a joint collaboration with Prof. Luis Nunes Vicente (Lehigh University) and Tommaso Giovannelli (University of Cincinnati). In this talk, we address stochastic optimization problems involving a composition of a non-smooth outer function and a smooth inner function, a formulation frequently encountered in machine learning and operations research. To deal with the non-differentiability of the outer function, we approximate the original non-smooth function using smoothing functions, which are continuously differentiable and approach the original function as a smoothing parameter goes to zero (at the price of increasingly higher Lipschitz constants). The proposed smoothing stochastic gradient method iteratively drives the smoothing parameter to zero at a designated rate. We establish convergence guarantees under strongly convex, convex, and nonconvex settings, proving convergence rates that match known results for non-smooth stochastic compositional optimization. In particular, for convex objectives, smoothing stochastic gradient achieves a~$1/T^{1/4}$ rate in terms of the number of stochastic gradient evaluations. We further show how the general compositional and finite sum compositional problems (widely used frameworks in large-scale machine learning and risk-averse optimization) fit the assumptions needed for the rates (unbiased gradient estimates, bounded second moments, and accurate smoothing errors). We will present numerical results indicating that smoothing stochastic gradient descent is a competitive method for certain classes of problems.
Talk 3: Forward to the Past: Optimal Stopping Problems are easy and Stopping Sets are optimal
Speaker: Daniel Ralph
Abstract: We use Poisson stopping times to study optimal stopping problems (eg, American options) and give a very general folk theorem: there is an optimal stopping set even for discontinuous data. For this function-space optimisation problem, we also show that (i) the stationary condition is a linear complementarity problem and can be solved by projective splitting, and (ii) its solution converges to solve the classic problem where you can stop at any time. (Joint work with R-J Lange and J van Casteren.)
