Welcome to Combinatorics and Optimization

Abstract:We consider the problem of minimizing the sum of a Lipschitz differentiable convex function and a proper closed convex function that admits efficient linear minimization oracles, subject to multiple smooth convex inequality constraints. We adapt the classical augmented Lagrangian (AL) method for these problems: in each iteration, our algorithm consists of one step of conditional gradient (CG) method applied to the AL function, followed by an update of the dual variable as in classical AL methods with a diminishing dual stepsize. We study the convergence rate of our algorithm under two standard stepsize rules for the CG method, namely, an open-loop stepsize and the short stepsize, and obtain a rate that matches the best-known complexity for this class of problems. We also establish accelerated rates when the aforementioned proper closed convex function is the indicator function of a uniformly convex set. This is a joint work with Xiaozhou Wang and Zev Woodstock.

Abstract:This talk presents a proximal stochastic subgradient method for minimizing the sum of an expected cost and a lower semicontinuous, prox-bounded function. We target a broad class of nonconvex integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, which in particular covers smooth losses with Lipschitz gradient. At each iteration, the expected cost is replaced by a sample average that is progressively refined, and the proximal stepsize is selected by an Armijo-type line search enforcing a
sufficient decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates more general problem formulations than existing methods and our analysis yields convergence guarantees that, to the best of our knowledge, are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded. Leveraging the Kurdyka-Lojasiewicz property, we further proof convergence of the whole trajectory to a single stationary point. Finally, for exponential-type desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to logarithmic factor, for both the function values and the iterates. This is a joint work with Felipe Atenas, Pedro Pérez-Aros and Alejandro
Jofré, from the University of Chile.

Abstract:

The $d$-dimensional Grothendieck constant is the smallest constant $K$ such that \begin{align*} \sup\left\{\sum_{i,j=1}^n A_{ij}\langle u_i,v_j\rangle:u_i,v_j\in S^{d-1}\right\}\le K\cdot\sup\left\{\sum_{i,j=1}^n A_{ij}x_iy_j:x_i,y_j\in\{-1,1\}\right\} \end{align*}for any $n\in\mathbb{N}$ and any real $n\times n$ matrix $A$. The inequality above, called the Grothendieck inequality, has made a deep impact in a variety of areas such as functional analysis, quantum information theory, and optimization. Determining the $d$-dimensional Grothendieck constant for any $d\ge 3$ is a long-standing open problem.

In this paper, we propose a worst operator in dimension 3, whose $\infty\to 1$ norm is conjectured to be $1/K_G(3)$. We study a related class of operators with nice geometric interpretations, and we prove the function $f:S^{d-1}\to\{-1,1\}$ corresponding to a hyperplane is uniquely optimal for this class, with the isoperimetric inequality lying at the heart of our proof.