Conjugate gradient methods are efficient for smooth optimization problems, while there are rare conjugate gradient based methods for solving a possibly nondifferentiable convex minimization problem. In this paper by making full use of inherent properties of Moreau-Yosida regularization and descent property of modified conjugate gradient method we propose a modified Fletcher-Reeves-type method f...

At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which ∂f(p) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f at p. We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semi...

Primal-dual path-following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed. Regularity properties of the path are proved. The path structure allows us to define approximating models, which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. The Moreau–Yosi...

With the increasing emphasis on supply chain vulnerabilities, effective mathematical tools for analyzing and understanding appropriate supply chain risk management are now attracting much attention. This paper presents a stochastic model of the multi-stage global supply chain network problem, incorporating a set of related risks, namely, supply, demand, exchange, and disruption. We provide a ne...

Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator B defined on an infinite-dimensional Hilbert space U is an admissible control operator if for every element u ∈ U the vector Bu defines an admissible control operator. The other conjecture says that B is an admissible control operator if a certain resolvent estimate ...

Moreau-Yosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to in nity or zero, respectively. In particular, the strong convergence of...

We develop a convergence theory called epi/hypo-convergence, for bivariate functions that essentially implies the convergence of their saddle points. We study the properties of this limiting process in particular. We characterize the limit functions associated to any collection of bivariate functions and obtain a compactness theorem for the space of saddle functions. Even when restricted to the...

We present several sequential exact Euclidean distance transform algorithms. The algorithms are based on fundamental transforms of convex analysis: The Legendre Conjugate or Legendre–Fenchel transform, and the Moreau envelope or Moreau-Yosida approximate. They combine the separability of the Euclidean distance with convex properties to achieve an optimal linear-time complexity. We compare them ...

This article aims to give an intuitive introduction to operator semi-groups and their generators from a probabilistic perspective. By ‘intuitive’ it is meant that the article relies mainly on heuristics and analogies to make its points. For example the definition of the generator is motivated from an analogy to the Cauchy functional equation g(t + s) = g(t)g(s) for real-valued functions. No pro...

We study a classification of the κ-times integrated semigroups (for κ > 0) by the (uniform) rate of convergence at the origin: ‖S(t)‖ = O(tα), 0 ≤ α ≤ κ. By an improved generation theorem we characterize this behaviour by Hille-Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize the convergence at the origin, as well as the existence of boun...