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Many least squares problems involve affine equality and inequality constraints. Although there are variety of methods for solving such problems, most statisticians find constrained estimation challenging. The current paper proposes a new…
This paper is concerned with a class of optimization problems with the nonnegative orthogonal constraint, in which the objective function is $L$-smooth on an open set containing the Stiefel manifold ${\rm St}(n,r)$. We derive a locally…
We present a general theory of exact penalty functions with vectorial (multidimensional) penalty parameter for optimization problems in infinite dimensional spaces. In comparison with the scalar case, the use of vectorial penalty parameters…
In this paper we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints, are locally smooth. For solving this problem, we propose a…
In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify…
In this paper, we focus on a class of constrained nonlinear optimization problems (NLP), where some of its equality constraints define a closed embedded submanifold $\mathcal{M}$ in $\mathbb{R}^n$. Although NLP can be solved directly by…
Our recent study (Lin and Ohtsuka, 2024) proposed a new penalty method for solving mathematical programming with complementarity constraints (MPCC). This method first reformulates MPCC as a parameterized nonlinear programming called gap…
In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relationships are imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity…
This paper considers stochastic optimization problems with weakly convex objective and constraint functions. We propose Prox-PEP, a proximal method equipped with quadratic subproblems. To handle nonlinear equality constraints, we employ an…
In this workshop, we present a compact but rigorous introduction to the basic language of nonlinear programming, variational inequalities, and complementarity systems. The goal is twofold. First, we explain the mathematical logic of…
Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with…
In exact sparse optimization problems on Rd (also known as sparsity constrained problems), one looks for solution that have few nonzero components. In this paper, we consider problems where sparsity is exactly measured either by the…
In the first part of this study, a convex-constrained penalized formulation was studied for a class of constant modulus (CM) problems. In particular, the error bound techniques were shown to play a vital role in providing exact penalization…
Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint…
The goal of this article is to study necessary and sufficient conditions for the exactness of penalty functions and the existence of global saddle points of augmented Lagrangians for well-posed (in a suitable sense) constrained optimization…
This study develops a framework for a class of constant modulus (CM) optimization problems, which covers binary constraints, discrete phase constraints, semi-orthogonal matrix constraints, non-negative semi-orthogonal matrix constraints,…
Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A…
We develop a general equality-constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970). Although it was historically considered to be computationally prohibitive in practice, we demonstrate…
In this article we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature. We discuss several approaches to the study of both locally and globally…
The last decade witnessed a rise in the importance of supervised learning applications involving {\em big data} and {\em big models}. Big data refers to situations where the amounts of training data available and needed causes difficulties…