arXiv subjects

Kaizhao Sun

Publications and source records attributed to Kaizhao Sun.

11 recordsLinked to original sources

arXiv2026

The Polarity Process for the Maximum-Volume Inscribed Ellipsoid Problem

We study the maximum-volume inscribed ellipsoid (MaxIE) problem for a polytope through a geometric iteration based on polarity. Given an interior point, the method forms the shifted polar polytope, computes its minimum-volume covering ellipsoid (MinCE), and then polarizes this covering ellipsoid back to obtain a new inscribed ellipsoid. This procedure was suggested by Khachiyan and Todd in 1993. Prior work studied basic polarity identities and gave an asymptotic-convergence argument for the exact process, but did not furnish a volume-ratio rate. Independently of that asymptotic argument, we develop a new analysis based on the log-volume of the polar MinCE as a potential function. We prove its convexity with an explicit gradient formula and establish global linear contraction of the potential gap along each trajectory. The contraction factor is existential and instance-dependent. This gives both a separate convergence proof and a finite volume-ratio iteration bound. We further analyze an inexact polarity process in which each MinCE subproblem is solved only approximately, and derive sufficient oracle tolerances for producing a prescribed volume approximation. Combining this outer analysis with existing algorithms for MinCE gives conditional arithmetic estimates based on the path-following Newton method, the barycentric coordinate descent, and the away-step Frank-Wolfe method. Numerical experiments compare these oracle choices with two MaxIE baselines and illustrate that performance depends on matching the MinCE solver to the instance geometry.

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arXiv2026

Automating Reformulation for Parallel ADMM

Many real-world optimization models contain exploitable sparsity and block structure, but this structure is often obscured in algebraic form, limiting the effectiveness of modern parallel algorithms. We propose an automatic pipeline that converts a generic multiblock problem into a canonical two-block formulation suitable for parallel Alternating Direction Method of Multipliers (ADMM). The method constructs a coupling graph, applies an edge-subdivision-based bipartization to obtain a bipartite representation, and produces an ADMM-ready decomposition with independent subproblems. Fast graph-traversal heuristics, a new mixed-integer linear program (MILP), and a learning-based graph neural network (GNN) surrogate model are developed to guide edge subdivision. Numerical experiments demonstrate that the resulting reformulations yield strong parallel ADMM performance. The entire pipeline is implemented in the open-source Julia package PDMO.jl.

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arXiv2026

ALiA: Adaptive Linearized ADMM

We propose ALiA, a novel adaptive variant of the alternating direction method of multipliers (ADMM). Specifically, ALiA is a variant of function-linearized proximal ADMM (FLiP ADMM), which generalizes the classical ADMM by leveraging the differentiable structure of the objective function, making it highly versatile. Notably, ALiA features an adaptive stepsize selection scheme that eliminates the need for backtracking linesearch. Motivated by recent advances in adaptive gradient and proximal methods, we establish point convergence of ALiA for convex and differentiable objectives. Furthermore, by introducing negligible computational overhead, we develop an alternative stepsize selection scheme for ALiA that improves the convergence speed both theoretically and empirically. Extensive numerical experiments on practical datasets confirm the accelerated performance of ALiA compared to standard FLiP ADMM. Additionally, we demonstrate that ALiA either outperforms or matches the practical performance of existing adaptive methods across problem classes where it is applicable.

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arXiv2024

Smoothed Proximal Lagrangian Method for Nonlinear Constrained Programs

This paper introduces a smoothed proximal Lagrangian method for minimizing a nonconvex smooth function over a convex domain with additional explicit convex nonlinear constraints. Two key features are 1) the proposed method is single-looped, and 2) an first-order iteration complexity of $\mathcal{O}(ε^{-2})$ is established under mild regularity assumptions. The first feature suggests the practical efficiency of the proposed method, while the second feature highlights its theoretical superiority. Numerical experiments on various problem scales demonstrate the advantages of the proposed method in terms of speed and solution quality.

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arXiv2024

Decomposition Methods for Global Solutions of Mixed-Integer Linear Programs

This paper introduces two decomposition-based methods for two-block mixed-integer linear programs (MILPs), which aim to take advantage of separable structures of the original problem by solving a sequence of lower-dimensional MILPs. The first method is based on the $\ell_1$-augmented Lagrangian method (ALM), and the second one is based on a modified alternating direction method of multipliers (ADMM). In the presence of certain block-angular structures, both methods create parallel subproblems in one block of variables, and add nonconvex cuts to update the other block; they converge to globally optimal solutions of the original MILP under proper conditions. Numerical experiments on three classes of MILPs demonstrate the advantages of the proposed methods on structured problems over the state-of-the-art MILP solvers.

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arXiv2023

Dual Descent ALM and ADMM

Classical primal-dual algorithms attempt to solve $\max_μ\min_{x} \mathcal{L}(x,μ)$ by alternatively minimizing over the primal variable $x$ through primal descent and maximizing the dual variable $μ$ through dual ascent. However, when $\mathcal{L}(x,μ)$ is highly nonconvex with complex constraints in $x$, the minimization over $x$ may not achieve global optimality, and hence the dual ascent step loses its valid intuition. This observation motivates us to propose a new class of primal-dual algorithms for nonconvex constrained optimization with the key feature to reverse dual ascent to a conceptually new dual descent, in a sense, elevating the dual variable to the same status as the primal variable. Surprisingly, this new dual scheme achieves some best iteration complexities for solving nonconvex optimization problems. In particular, when the dual descent step is scaled by a fractional constant, we name it scaled dual descent (SDD), otherwise, unscaled dual descent (UDD). For nonconvex multiblock optimization with nonlinear equality constraints, we propose SDD-ADMM and show that it finds an $ε$-stationary solution in $\mathcal{O}(ε^{-4})$ iterations. The complexity is further improved to $\mathcal{O}(ε^{-3})$ and $\mathcal{O}(ε^{-2})$ under proper conditions. We also propose UDD-ALM, combining UDD with ALM, for weakly convex minimization over affine constraints. We show that UDD-ALM finds an $ε$-stationary solution in $\mathcal{O}(ε^{-2})$ iterations. These complexity bounds for both algorithms either achieve or improve the best-known results in the ADMM and ALM literature. Moreover, SDD-ADMM addresses a long-standing limitation of existing ADMM frameworks.

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arXiv2023

An ADMM-based Distributed Optimization Method for Solving Security-Constrained AC Optimal Power Flow

In this paper, we study efficient and robust computational methods for solving the security-constrained alternating current optimal power flow (SC-ACOPF) problem, a two-stage nonlinear optimization problem with disjunctive constraints, that is central to the operation of electric power grids. The first-stage problem in SC-ACOPF determines the operation of the power grid in normal condition, while the second-stage problem responds to various contingencies of losing generators, transmission lines, and transformers. The two stages are coupled through disjunctive constraints, which model generators' active and reactive power output changes responding to system-wide active power imbalance and voltage deviations after contingencies. Real-world SC-ACOPF problems may involve power grids with more than 30k buses and 22k contingencies and need to be solved within 10-45 minutes to get a base case solution with high feasibility and reasonably good generation cost. We develop a comprehensive algorithmic framework to solve SC-ACOPF that meets the challenge of speed, solution quality, and computation robustness. In particular, we develop a smoothing technique to approximate disjunctive constraints into a smooth structure which can be handled by interior-point solvers; we design a distributed optimization algorithm to efficiently generate first-stage solutions; we propose a screening procedure to prioritize contingencies; and finally, we develop a reliable and parallel architecture that integrates all algorithmic components. Extensive tests on industry-scale systems demonstrate the superior performance of the proposed algorithms.

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arXiv2022

A two-level distributed algorithm for nonconvex constrained optimization

This paper aims to develop distributed algorithms for nonconvex optimization problems with complicated constraints associated with a network. The network can be a physical one, such as an electric power network, where the constraints are nonlinear power flow equations, or an abstract one that represents constraint couplings between decision variables of different agents. Despite the recent development of distributed algorithms for nonconvex programs, highly complicated constraints still pose a significant challenge in theory and practice. We first identify some difficulties with the existing algorithms based on the alternating direction method of multipliers (ADMM) for dealing with such problems. We then propose a reformulation that enables us to design a two-level algorithm, which embeds a specially structured three-block ADMM at the inner level in an augmented Lagrangian method (ALM) framework. Furthermore, we prove the global and local convergence as well as iteration complexity of this new scheme for general nonconvex constrained programs, and show that our analysis can be extended to handle more complicated multi-block inner-level problems. Finally, we demonstrate with computation that the new scheme provides convergent and parallelizable algorithms for various nonconvex applications, and is able to complement the performance of the state-of-the-art distributed algorithms in practice by achieving either faster convergence in optimality gap or in feasibility or both.

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arXiv2022

Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. We then develop four algorithms for solving DC programs with and without linear constraints based on the DME smoothing. In particular, for a smoothed DC program without linear constraints, we show that the classic gradient descent method as well as an inexact variant can obtain a stationary solution in the limit with a convergence rate of $\mathcal{O}(K^{-1/2})$, where $K$ is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named LCDC-ALM and composite LCDC-ALM, focusing on different structures of the DC objective function. We show that both algorithms find an $ε$-approximate stationary solution of the original DC program in $\mathcal{O}(ε^{-2})$ iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms.

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arXiv2022

Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition

The optimal power flow problem is central to many tasks in the design and operation of electric power grids. This problem seeks the minimum cost operating point for an electric power grid while satisfying both engineering requirements and physical laws describing how power flows through the electric network. By additionally considering the possibility of component failures and using an accurate AC power flow model of the electric network, the security-constrained AC optimal power flow (SC-AC-OPF) problem is of paramount practical relevance. To assess recent progress in solution algorithms for SC-AC-OPF problems and spur new innovations, the U.S. Department of Energy's Advanced Research Projects Agency--Energy (ARPA-E) organized Challenge 1 of the Grid Optimization (GO) competition. This paper describes the SC-AC-OPF problem formulation used in the competition, overviews historical developments and the state of the art in SC-AC-OPF algorithms, discusses the competition, and summarizes the algorithms used by the top three teams in Challenge 1 of the GO Competition (Teams gollnlp, GO-SNIP, and GMI-GO).

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arXiv2021

A Two-level ADMM Algorithm for AC OPF with Global Convergence Guarantees

This paper proposes a two-level distributed algorithmic framework for solving the AC optimal power flow (OPF) problem with convergence guarantees. The presence of highly nonconvex constraints in OPF poses significant challenges to distributed algorithms based on the alternating direction method of multipliers (ADMM). In particular, convergence is not provably guaranteed for nonconvex network optimization problems like AC OPF. In order to overcome this difficulty, we propose a new distributed reformulation for AC OPF and a two-level ADMM algorithm that goes beyond the standard framework of ADMM. We establish the global convergence and iteration complexity of the proposed algorithm under mild assumptions. Extensive numerical experiments over some largest test cases from NESTA and PGLib-OPF (up to 30,000-bus systems) demonstrate advantages of the proposed algorithm over existing ADMM variants in terms of convergence, scalability, and robustness. Moreover, under appropriate parallel implementation, the proposed algorithm exhibits fast convergence comparable to or even better than the state-of-the-art centralized solver.

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