In derivative-free optimization (DFO), one minimizes functions for which the gradient is unavailable or expensive to compute. In many applications, objective function values and gradients are noisy due to simulations or system randomness. A class of standard direct-search methods for DFO accept a trial point when it decreases the objective function by an amount proportional … Read more
This document provides an extended and rigorous framework dedicated to the geometric analysis of the 3D incompressible Navier-Stokes equations \cite{ESS2003}. We comprehensively develop geometric proofs related to decay estimates, blow-up profiles, and the foundational partial regularity theory of Caffarelli, Kohn, and Nirenberg (CKN). We examine in detail the Hausdorff dimension of potential singular sets, local … Read more
Self-intersection is a fundamental feasibility constraint in shape optimization: a self-crossing boundary leaves its interior, normal field, and finite-element mesh ill-defined, yet most existing barriers rely on heuristic geometric-proximity measures rather than certifying self-intersection directly. We propose a self-intersection barrier grounded in algebraic detection. We derived two bivariate polynomials from a curve’s Fourier coefficients whose … Read more
In this paper, we study nonconvex stochastic optimization with {decision-dependent distributions}, where the decision variable influences the underlying sampling distribution and only stochastic function-value feedback is available. We address two challenges {induced by decision-dependent distributions}: transport error in momentum-based gradient tracking and variance inflation in zeroth-order estimation. We first develop a Polyak-momentum zeroth-order method that … Read more
The ESA/NASA joint LISA (laser interferometer space antenna) mission is designed to detect gravitational waves to perform gravitational astronomy. A key mission requirement is the maintenance of a three-spacecraft constellation in a near-equilateral triangular configuration with a prescribed inter-spacecraft separation. Existing approaches have addressed this problem using simplified dynamical models to enhance tractability; however, the … Read more
In this paper, we propose new inexact adaptive proximal gradient algorithms for solving nonconvex composite optimization problems, where the objective is the sum of a differentiable nonconvex function and a convex non-differentiable function. A new relative error criterion to compute the proximal operator inexactly has been proposed together with new adaptive strate- gies for selecting … Read more
We present an inexact Augmented Lagrangian algorithm for solving nonlinear, non-convex optimization problems. Unlike most recently proposed Augmented Lagrangian methods with worst-case complexity guarantees, we utilize adaptive penalty parameter updates and full dual stepsizes. We show that the method matches the best known worst-case complexity results for Augmented Lagrangian methods (up to logarithmic factors) when … Read more
We systematically investigate the potentials of improving the lower bound obtained with a linear relaxation of the Quadratic Unconstrained Binary Optimization problem by integrating curvature information from an accompanying quadratic convex underestimator via gradient inequalities. On the one hand, we exemplify to which extent this hybrid approach may provide a lower bound that is strictly … Read more
Path planning in environments containing obstacles has numerous practical applications. The problem is challenging because it is inherently nonlinear and nonconvex. Consequently, a variety of techniques have been developed to address this problem, among which machine learning and optimal control (or optimization) have emerged as two prominent approaches. In general, machine learning methods do not … Read more
This paper proposes a new arc-search interior-point algorithm for the nonlinear constrained optimization problem. The proposed algorithm uses the second-order derivatives to construct a search arc that approaches the optimizer. Because the arc stays in the interior set longer than any straight line, it is expected that the scheme will generate a better new iterate … Read more