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Bilevel optimization is an important class of optimization problems where one optimization problem is nested within another. While various methods have emerged to address unconstrained general bilevel optimization problems, there has been a…
This paper addresses a class of (non-)convex optimization problems subject to general convex constraints, which pose significant challenges for traditional methods due to their inherent non-convexity and diversity. Conventional convex…
This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problem in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo linesearch…
A quadratically constrained quadratic programming problem is considered in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are provided to guarantee that the…
Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations…
The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to…
We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation…
Nonconvex sparse models have received significant attention in high-dimensional machine learning. In this paper, we study a new model consisting of a general convex or nonconvex objectives and a variety of continuous nonconvex…
A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented. The proposed method is based on a recursively feasible and descending sequential convex programming procedure proven…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of…
We propose algorithms and software for computing projections onto the intersection of multiple convex and non-convex constraint sets. The software package, called SetIntersectionProjection, is intended for the regularization of inverse…
The convex feasibility problem asks to find a point in the intersection of a collection of nonempty closed convex sets. This problem is of basic importance in mathematics and the physical sciences, and projection (or splitting) methods…
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately…
In this paper we introduce a technique to produce tighter cutting planes for mixed-integer non-linear programs. Usually, a cutting plane is generated to cut off a specific infeasible point. The underlying idea is to use the infeasible point…
It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear…
This paper considers a conceptual version of a convex optimization algorithm whic is based on replacing a convex optimization problem with the root-finding problem for the approximate sub-differential mapping which is solved by repeated…
In this article we consider a consistent convex feasibility problem in a real Hilbert space defined by a finite family of sets $C_i$. We are interested, in particular, in the case where for each $i$, $C_i=Fix (U_i)=\{z\in \mathcal H\mid…
We consider solving large scale nonconvex optimisation problems with nonnegativity constraints. Such problems arise frequently in machine learning, such as nonnegative least-squares, nonnegative matrix factorisation, as well as problems…
This work is devoted to establish the strong convergence results of an iterative algorithm generated by the shrinking projection method in Hilbert spaces. The proposed approximation sequence is used to find a common element in the set of…