This work introduces an inexact cubic regularization method with adaptive reuse of Hessian approximations to solve general non-convex optimization problems. In the proposed approach, the gradient is computed inexactly and updated at every iteration, whereas the Hessian approximation is updated at a specific iteration and then reused for $m$ subsequent iterations (a lazy strategy), where … Read more
We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption … Read more
In this study, we model the supervised feature selection problem using a novel approach: convex bi-objective optimization. Traditional methods have addressed this problem by maximizing relevance to class labels and minimizing redundancy among features. Recently, Wang et al. [30] formulated this problem as a single-objective convex optimization, yielding only a unique solution. Unlike that, we … Read more
Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct nonlinear formulations, we consider packing circles in squares and fixed-perimeter rectangles, packing circles into minimum-area ellipses, packing regular polygons into regular polygons, … Read more
We consider unconstrained derivative-free optimization problems in which only inexact function evaluations are available. Specifically, we study the setting where the oracle returns function values with partially controllable inexactness, with the error bounded linearly by a user-specified accuracy parameter, but with an unknown proportionality constant. This framework captures optimization problems arising from approximate simulations or … Read more
The inexact adaptive stepsizes for the conjugate gradient method and the quasi-Newton method are very rare. The exact stepsizes in the gradient method, the conjugate gradient method and the quasi-Newton method for strictly convex quadratic optimization have a unified framework, while the unified framework for inexact adaptive stepsizes in the gradient method, the conjugate gradient … Read more
Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a computable optimality certificate. We first analyze primal and dual methods based … Read more
In this work we develop and analyze a semi-smooth Newton method for the general non- linear conic programming problem. In particular, we study the problem with a generalized simplicial cone, i.e., the image of a symmetric cone under a linear mapping. We generalize Robinson’s normal equations to a conic setting, yielding what we call the … Read more
A unified framework for first-order optimization algorithms for nonconvex unconstrained optimization is proposed that uses adaptively preconditioned gradients and includes popular methods such as full and diagonal AdaGrad, AdaNorm, as well as an adpative variants of Muon. This framework also allows combining heterogeneous geometries across different groups of variables while preserving a unified convergence analysis. … Read more
In this paper, we consider nonlinear optimization problems with a stochastic objective function and deterministic equality constraints. We propose an inexact two-stepsize stochastic sequential quadratic programming (SQP) algorithm and analyze its worst-case complexity under mild assumptions. The method utilizes a step decomposition strategy and handles stochastic gradient estimates by assigning different stepsizes to different components … Read more