2022 Optimization Days
HEC Montréal, Québec, Canada, 16 — 18 May 2022
WC5  Numerical optimization and linear algebra with Julia IV
May 18, 2022 03:30 PM – 05:10 PM
Location: LouisLaberge (red)
Chaired by Victor Zavala
4 Presentations

03:30 PM  03:55 PM
On the computation of the Bdifferential of the Min Cfunction for the balanced linear complementarity problem
A complementarity problem is often dealt with Cfunctions, allowing
us to reformulate the problem as a nonsmooth system of equations.
Among them, the minimum function has interesting properties,
including its simplicity. The semismooth Newton method makes use
of the Bdifferential of this function. In this talk, we address
the problem of describing completely this Bdifferential
for a balanced linear complementarity problem.Along the way, we discuss equivalent forms of this problem, such as
enumerating the sectors formed by an arrangement of hyperplanes
all containing the origin, listing the biconvex bipartitions of a finite set
and ascertaining all the signed solvability of systems of linear inequalities.After giving several properties of the Bdifferential, we propose a
numerical technique to describe it. Focusing on the last equivalent form,
we implement an incremental approach, with overall complexity
better than the brute force technique, thus often much faster in practice.
At each step, several optimization problems with a linear cost
and a ball constraint must be solved. The efficiency of their solution
computation is grounded on a selfconcordant function
that describes the feasible set. Several approaches are implemented
to solve the optimization problems. Numerical results are presented. 
03:55 PM  04:20 PM
Clustering tensorvariate data using mixture models implemented in the Julia programming language
Contemporary scientific data has become increasingly complex, taking a multitude of different forms. One such form is tensors or multidimensional arrays. Using tensors in clustering and classification problems can be challenging due to the high dimensionality of each tensor in a sample and the computational resources required to estimate model parameters. Motivated by our work in pediatric research, a tensorvariate normal mixture model is proposed to cluster fiveway data. Fiveway data consists of a sample of mode four tensors, one per subject. The model has interpretable parameters for the mean tensor and the covariance matrices for each mode of the array. The model parameters are estimated using an EM algorithm. Our model is implemented in Julia, a dynamic and highperformance numerical computing language.

04:20 PM  04:45 PM
Progress and applications of a FrankWolfe solver
FrankWolfe algorithms are firstorder methods that support structured convex constraints, over which linear optimization can be performed efficiently. We present FrankWolfe.jl, a library for flexible and highperformance FrankWolfe algorithms. In particular, we will detail design choices that allow warm starts, scaling to large problems and some use cases leveraging those features.

04:45 PM  05:10 PM
Exponential Decay of Sensitivity in GraphStructured Nonlinear Programs
We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that for a given pair of nodes, the sensitivity of the primaldual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays exponentially as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under fairly standard assumptions: the strong secondorder sufficiency condition and the linear independence constraint qualification. We discuss how this property provides new and interesting insights on the behavior of complex systems and how it enables the design of new and scalable decomposition algorithms and software.