Optimization of Linear System under Convex Endpoint Constraints

Structured Convex Optimization under Submodular Constraints

A number of discrete and continuous optimization problems in machine learning are related to convex minimization problems under submodular constraints. In this paper, we deal with a submodular function with a directed graph structure, and we show that a wide range of convex optimization problems under submodular constraints can be solved much more efficiently than general submodular optimizatio...

Convex programming under a linear optimization oracle The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle

This paper considers a general class of iterative optimization algorithms, referred to as linearoptimization-based convex programming (LCP) methods, for solving large-scale convex programming (CP) problems. The LCP methods, covering the classic conditional gradient (CndG) method (a.k.a., Frank-Wolfe method) as a special case, can only solve a linear optimization subproblem at each iteration. In...

Maximization of A convex quadratic function under linear constraints

Separable convex optimization problems with linear ascending constraints

Separable convex optimization problems with linear ascending inequality and equality constraints are addressed in this paper. Under an ordering condition on the slopes of the functions at the origin, an algorithm that determines the optimum point in a finite number of steps is described. The optimum value is shown to be monotone with respect to a partial order on the constraint parameters. More...

Iteratively Solving Linear Inverse Problems under General Convex Constraints

We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In cer...