Structure, Sparsity, and Communication in Modern Convex Optimization
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ye-ghye-phd-eecs-2026-thesis.pdf
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Author(s)
Ye, Guanghao
Advisor(s)
Kelner, Jonathan A.
Date Issued
February 2026
Publisher
Massachusetts Institute of Technology
Abstract
This thesis develops efficient algorithms for convex optimization by exploiting problem structure. While general-purpose methods offer broad applicability, they often fail to leverage the sparsity, decomposability, and communication constraints present in concrete instances. We show that systematically using such structure yields substantially improved complexity bounds in three settings: graph-structured optimization, non-smooth Lipschitz optimization, and distributed convex optimization. The first part presents nearly-linear time algorithms for optimization on structured graphs. We resolve a decades-old open problem by giving the first nearly-linear time algorithm for minimum-cost flow on planar graphs. Our approach combines interior point methods with dynamic data structures based on nested dissection, exploiting planar separators to achieve sublinear amortized time per iteration. These techniques extend more broadly: we obtain faster algorithms for minimum-cost flow and linear programming on any graph class with small balanced separators, including separable graphs and graphs of bounded treewidth. The second part develops algorithms for non-smooth Lipschitz optimization that exploit decomposable structure. We consider minimizing a sum of convex Lipschitz functions where each function depends on a sparse subset of coordinates. Defining the total effective dimension as the sum of these individual dimensions, we obtain nearly-linear gradient oracle complexity in this parameter, matching information-theoretic lower bounds up to logarithmic factors. Our approach combines cutting-plane and interior point methods, maintaining inner and outer approximations that converge around the optimum. We also derive new complexity guarantees for gradient sampling methods on general, possibly non-convex, Lipschitz functions. The third part studies the bit complexity of distributed convex optimization. When data is partitioned across machines, the total number of bits communicated is a key measure of practical efficiency under finite-precision arithmetic. We design communication-efficient algorithms for least squares regression, low-rank approximation, linear programming, and decomposable function minimization. For regression, we show that obtaining a constant-factor approximation requires no more communication than solving consistent linear systems. For linear programming, we reduce bit complexity from cubic to nearly-linear in the dimension, and we complement these upper bounds with nearly matching lower bounds. A unifying theme is the interplay between discrete and continuous techniques: graph separators and data structures from combinatorial algorithms combine with interior point and cutting-plane methods from continuous optimization to yield complexity improvements that neither approach attains in isolation.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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