Jonathan M. Borwein

On Projection Algorithms for Solving Convex Feasibility Problems

Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

Proper Efficient Points for Maximizations with Respect to Cones

Proper efficient points (Pareto maxima) are defined in tangent cone terms and are characterized by the existence of equivalent real-valued maximization problems.

Partially finite convex programming, part I: quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.

On the convergence of von Neumann's alternating projection algorithm for two sets

We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejér monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.

A cubic counterpart of Jacobi's identity and the AGM

We produce exact cubic analogues of Jacobis celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is

Duality relationships for entropy-like minimization problems

a_n+1 := a_n + 2b_n / 3

Super efficiency in vector optimization

and b_n+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation.

Stability and regular points of inequality systems

This paper considers the minimization of a convex integral functional over the positive cone of an

Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces

L_p

Bregman Monotone Optimization Algorithms

space, subject to a finite number of linear equality constraints. Such problems arise in spectral estimation, where the bjective function is often entropy-like, and in constrained approximation. The Lagrangian dual problem is finite-dimensional and unconstrained. Under a quasi-interior constraint qualification, the primal and dual values are equal, with dual attainment. Examples show the primal value may not be attained. Conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum. These conditions are satisfied in many examples.