arXiv subjects

Zev Woodstock

Publications and source records attributed to Zev Woodstock.

8 recordsLinked to original sources

arXiv2026

Frank-Wolfe with Moreau Envelope Smoothing for Nonsmooth Nonconvex Problems

We present and analyze Frank-Wolfe with Moreau Envelope Smoothing (FRAMES) for solving nonsmooth nonconvex constrained optimization problems, taking advantage of iterative smoothing via the Moreau envelope followed by one Frank-Wolfe step per iteration. The problem template we consider encompasses splitting problems with multiple convex constraint sets as well as problems with nonsmooth weakly convex regularizers like the MCP or SCAD penalties. We prove convergence, with rates, for both of these cases under a variety of mild assumptions, including inconsistent problems. Additionally, we highlight a new relationship between the Frank-Wolfe gap for a problem with nonsmooth objective and the Frank-Wolfe gap for a smoothed surrogate problem, demonstrating suboptimality of prior analyses. Numerical experiments are performed for matrix factorization problems and nonconvex quadratic splitting over multiple convex constraint sets, where the improvements in analysis are empirically observed.

math.OC↗

arXiv2026

A conditional-gradient-based single-loop augmented Lagrangian method for inequality constrained optimization

We consider the problem of minimizing the sum of a Lipschitz differentiable convex function $f$ and a proper closed convex function $h$ 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 the 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 convergence rate that matches the best-known complexity for this class of problems. We also establish accelerated rates when $h$ is the indicator function of a uniformly convex set.

math.OC↗

arXiv2025

Flexible block-iterative analysis for the Frank-Wolfe algorithm

We prove that the block-coordinate Frank-Wolfe (BCFW) algorithm converges with state-of-the-art rates in both convex and nonconvex settings under a very mild "block-iterative" assumption. This appears to be the first result on BCFW addressing the setting of nonconvex objective functions with Lipschitz-continuous gradients and no additional assumptions. This analysis newly allows for (I) progress without activating the most-expensive linear minimization oracle(s), LMO(s), at every iteration, (II) parallelized updates that do not require all LMOs, and therefore (III) deterministic parallel update strategies that take into account the numerical cost of the problem's LMOs. Our results apply for short-step BCFW as well as an adaptive method for convex functions. New relationships between updated coordinates and primal progress are proven, and a favorable speedup is demonstrated using FrankWolfe.jl.

math.OC↗

arXiv2025

High-precision linear minimization is no slower than projection

This note demonstrates that, for all compact convex sets, high-precision linear minimization can be performed via a single evaluation of the projection and a scalar-vector multiplication. In consequence, if $\varepsilon$-approximate linear minimization takes at least $L(\varepsilon)$ real vector-arithmetic operations and projection requires $P$ operations, then $\mathcal{O}(P)\geq \mathcal{O}(L(\varepsilon))$ is guaranteed. This concept is expounded with examples, an explicit error bound, and an exact linear minimization result for polyhedral sets.

math.OC↗

arXiv2024

Splitting the Conditional Gradient Algorithm

We propose a novel generalization of the conditional gradient (CG / Frank-Wolfe) algorithm for minimizing a smooth function $f$ under an intersection of compact convex sets, using a first-order oracle for $\nabla f$ and linear minimization oracles (LMOs) for the individual sets. Although this computational framework presents many advantages, there are only a small number of algorithms which require one LMO evaluation per set per iteration; furthermore, these algorithms require $f$ to be convex. Our algorithm appears to be the first in this class which is proven to also converge in the nonconvex setting. Our approach combines a penalty method and a product-space relaxation. We show that one conditional gradient step is a sufficient subroutine for our penalty method to converge, and we provide several analytical results on the product-space relaxation's properties and connections to other problems in optimization. We prove that our average Frank-Wolfe gap converges at a rate of $\mathcal{O}(\ln t/\sqrt{t})$, -- only a log factor worse than the vanilla CG algorithm with one set.

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arXiv2023

On a Frank-Wolfe Approach for Abs-smooth Functions

We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our problem setting is motivated by various applications that lead to nonsmoothness, such as $\ell_1$ regularization, phase retrieval problems, or ReLU activation in machine learning. To handle the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is {\em identical} to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.

math.OC↗

arXiv2016

Obstructions to convexity in neural codes

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstructions. Curto et al. proved the converse for all neural codes on at most four neurons. Via a counterexample on five neurons, we show that this converse is false in general. Additionally, we classify all codes on five neurons with no local obstructions. This classification is enabled by our enumeration of connected simplicial complexes on 5 vertices up to isomorphism. Finally, we examine how local obstructions are related to maximal codewords (maximal sets of neurons that co-fire). Curto et al. proved that a code has no local obstructions if and only if it contains certain "mandatory" intersections of maximal codewords. We give a new criterion for an intersection of maximal codewords to be non-mandatory, and prove that it classifies all such non-mandatory codewords for codes on up to 5 neurons.

q-bio.NC↗

arXiv2016

Analyzing Multistationarity in Chemical Reaction Networks using the Determinant Optimization Method

Multistationary chemical reaction networks are of interest to scientists and mathematicians alike. While some criteria for multistationarity exist, obtaining explicit reaction rates and steady states that exhibit multistationarity for a given network -- in order to check nondegeneracy or determine stability of the steady states, for instance -- is nontrivial. Nonetheless, we accomplish this task for a certain family of sequestration networks. Additionally, our results allow us to prove the existence of nondegenerate steady states for some of these sequestration networks, thereby resolving a subcase of a conjecture of Joshi and Shiu. Our work relies on the determinant optimization method, developed by Craciun and Feinberg, for asserting that certain networks are multistationary. More precisely, we implement the construction of reaction rates and multiple steady states which appears in the proofs that underlie their method. Furthermore, we describe in detail the steps of this construction so that other researchers can more easily obtain, as we did, multistationary rates and steady states.

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