Related papers: Convex duality in stochastic programming and mathe…
This article studies convex duality in stochastic optimization over finite discrete-time. The first part of the paper gives general conditions that yield explicit expressions for the dual objective in many applications in operations…
This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex…
This article characterizes conjugates and subdifferentials of convex integral functionals over linear spaces of cadlag stochastic processes. The approach is based on new measurability results on the Skorokhod space and new interchange rules…
This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows…
Geometric duality theory for multiple objective linear programming problems turned out to be very useful for the development of efficient algorithms to generate or approximate the whole set of nondominated points in the outcome space. This…
A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the…
We show that a broad range of convex optimization algorithms, including alternating projection, operator splitting, and multiplier methods, can be systematically derived from the framework of subspace correction methods via convex duality.…
We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in Machine Learning applications such as structured support vector machines, matrix factorization and min-min or…
This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in 1976. We derive an explicit dual problem in terms of two dual variables, one of which is the…
We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our…
This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but…
This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality…
Hidden convexity is a powerful idea in optimization: under the right transformations, nonconvex problems that are seemingly intractable can be solved efficiently using convex optimization. We introduce the notion of a Lagrangian dual…
In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…
We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal…
Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by Freund (1987}, seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be…