Spencer C. Schantz Technical Talk

Speaker: Yurii Nesterov, University of Louvain

Title: Inexact High-Order Proximal-Point Methods with Auxiliary Search Procedure

Date: Monday, August 2, 2021


The ISE Department was honored to have Professor Yurii Nesterov give a Spencer C. Schantz Technical Talk via Zoom during the MOPTA 2021 Conference on Monday, August 2, 2021.

You may read Professor Nesterov’s abstract and more about his impressive career below.


In this paper, we present the further development of Bi-Level Unconstrained Minimization by a new pth-order proximal-point method with the convergence rate O(1=k(1+3p)=2), where k is the iteration counter. In this method, we replace the auxiliary line-search procedure by a segment search. This allows bounding its complexity by a logarithm of the desired accuracy. Each step in this search needs an approximate computation of a high-order proximal-point operator. Under assumption on the boundedness of the (p+1)th derivative of the objective function, this can be done by one step of the pth order augmented tensor method. In this way, for p = 2, we get a new second-order method with the rate of convergence O(1=k7=2) and logarithmic complexity of the auxiliary search at each iteration. Another possibility is to compute the proximal-point operator by a lower-order minimization method. As an example, for p = 3, we consider the upper-level process convergent as O(1=k5). Assuming boundedness of the fourth derivative, an appropriate approximation of the proximal-point operator can be computed by a second-order method in a logarithmic number of iterations. This combination gives a second-order scheme with much better complexity than the existing theoretical limits.


BIOGRAPHY. Born: 1956, Moscow. Master degree 1977, Moscow State University. Doctor degree 1984. Professor at Center for Operations Research and Econometrics, UC Louvain, Belgium. Author of 5 monographs and more than 120 refereed papers in the leading optimization journals. International recognition: Dantzig Prize 2000, John von Neumann Theory Prize 2009, Charles Broyden prize 2010, Francqui Chair (Liege University 2011-2012), SIAM Outstanding paper award (2014), EURO Gold Medal 2016. Main research direction is the development of efficient numerical methods for convex and nonconvex optimization problems supported by the global complexity analysis: general interior-point methods (theory of self-concordant functions), fast gradient methods (smoothing technique), global complexity analysis of second-order and tensor schemes (cubic regularization of the Newton’s method), accelerated proximal-point methods.

This lecture series is endowed in the name of the late Spencer C. Schantz, who graduated from Lehigh in 1955 with a B.S. in Industrial Engineering. Following progressive responsibilities with several electrical manufacturing companies, in 1969 he founded U.S. Controls Corporation and became its first CEO and President.

The Spencer C. Schantz Distinguished Lecture Series was established by his wife Jerelyn as a valuable educational experience for faculty, students and friends of Lehigh’s Industrial and Systems Engineering department.