Related papers: First-Order Optimality Conditions for Mathematical…

The well known constant rank constraint qualification [Math. Program. Study 21:110--126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the…

Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second-…

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the…

This paper explores some sufficient conditions for the enhanced solvability of strong vector equilibrium problems, which can be established via a variational approach. Enhanced solvability here means existence of solutions, which are strong…

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…

For the rank regularized minimization problem, we introduce several kinds of stationary points by the problem itself and its equivalent reformulations including the mathematical program with an equilibrium constraint (MPEC), the global…

First-order conic optimization solvers are sensitive to problem conditioning and typically perform poorly in the face of ill-conditioned problem data. To mitigate this, we propose an approach to preconditioning--the hypersphere…

This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with…

Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate…

Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these…

We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic…

Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show for any Lipschitz, weakly convex objectives and…

This thesis focuses on developing and analyzing accelerated and inexact first-order methods for solving or finding stationary points of various nonconvex composite optimization (NCO) problems. The main tools mainly come from variational and…

A framework is developed for applying accelerated methods to general hyperbolic programming, including linear, second-order cone, and semidefinite programming as special cases. The approach replaces a hyperbolic program with a convex…

In this paper, we study nonconvex constrained optimization problems with both equality and inequality constraints, covering deterministic and stochastic settings. We propose a novel first-order algorithm framework that employs a…

In this paper, we investigate second-order necessary conditions and exact penalty of mathematical programs with switching constraints (MPSC). Some new second-order constraint qualifications and second-order quasi-normality are introduced…

This paper is an attempt to remedy the problem of slow convergence for first-order numerical algorithms by proposing an adaptive conditioning heuristic. First, we propose a parallelizable numerical algorithm that is capable of solving…

We present exact mixed-integer linear programming formulations for verifying the performance of first-order methods for parametric quadratic optimization. We formulate the verification problem as a mixed-integer linear program where the…

This paper proposes a constrained maximum likelihood estimator for sequential search models, using the MPEC (Mathematical Programming with Equilibrium Constraints) approach. This method enhances numerical accuracy while avoiding ad hoc…

It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality…