Abstracts

The Proximal-proximal Gradient Algorithm

Ting Kei Pong

Department of Mathematics, UBC

Abstract

We consider minimizing the sum of a smooth convex function, with Lipschitz continuous gradient, and a nonsmooth part, which is a composition of a proper closed convex function P and a nonzero linear map. Even under the assumption that the proximal mapping of P is easy to compute, a direct application of the widely used proximal gradient algorithm does not necessarily lead to easy subproblems. In this talk, we present a new algorithm, the proximal-proximal gradient algorithm, which admits easy subproblems. Our algorithm reduces to the proximal gradient algorithm if the linear map is just the identity map, and can be viewed as a "very inexact" inexact proximal gradient algorithm. We show that the whole sequence generated from the algorithm converges to an optimal solution, and establish an upper bound on iteration complexity.

New representation theorems for completely monotone and Bernstein functions with a convexity type condition on their measures

Hristo Sendov

Department of Statistical and Actuarial Sciences, University of Western Ontario

Abstract

Completely monotone and Bernstein functions are classical objects finding wide ranging applications from the theoretical, such as operator monotone functions, to the applied, such as Levy processes. Many classes of Bernstein functions have been singled out and their properties investigated, for example complete, special, and Thorin Bernstein functions. We introduce yet another class of completely monotone and Bernstein functions defined via convexity properties of their representing measures. We derive a representation theorem for Bernstein functions with a harmonic concavity condition on its Levy measure. It is a variant of the well-known Levy-Khintchine representation theorem. We characterize harmonic convexity (concavity) of the representing measures of completely monotone and Bernstein functions in terms of the functions themselves. This is a joint work with Shen Shan.

Full Stability in Finite-Dimensional Optimization

Boris Mordukhovich

Department of Mathematics, Wayne State University

Abstract

The talk is devoted to full stability of optimal solutions in general settings of finite-dimensional optimization with applications to particular models of constrained optimization problems including those of conic and specifically semidefinite programming. Developing a new technique of variational analysis and generalized differentiation, we derive second-order characterizations of full stability, in both Lipschitzian and Holderian settings, and establish their relationships with the conventional notions of strong regularity and strong stability for a large class of problems of constrained optimization with twice continuously differentiable data.

[Based on the joint work with Tran Nghia and Terry Rockafellar.]

Lipschitzian and Holderian Stability of parametric variational conditions

Nghia Tran

Department of Mathematics, UBC Okanagan

Abstract

In this talk we provide several new sufficient conditions for existence of a locally unique and Lipschitz (or Holder) solution of a parametric variational condition via second-order generalized differentiation. In the framework of variational conditions over parametric nonlinear constraints, our conditions become independent of the well-known strong coherent orientation condition and strictly weaker than the classical strong second-order sufficient condition under the constant rank constraint qualification.

[Based on the joint work with B. S. Mordukhovich]

Smoothing Techniques and Nesterov Accelerated Gradient Method for Location Problems

Mau Nam Nguyen

Fariborz Maseeh Department of Mathematics and Statistis, Portland State University

Abstract

Many recent applications require solving optimization problems with nonsmooth data. The idea to approximate nonsmooth optimization problems by smooth ones is a natural approach. Our goal in this talk is to apply a number of smoothing techniques to solve location problems that involve distances generated by different norms and distances to convex sets. We present effective numerical algorithms for solving these problems. Results on convergence of the algorithms and preliminary numerical tests are reported.

Convexity and Duality in the Economics of Financial Equilibrium

Terry Rockafellar

Department of Mathematics, University of Washington

Abstract

In classical models of economic equilibrium with markets of goods, agents maximize quasi-concave utility subject to budget constraints based on the value of their initial holdings. When financial markets are added, there are serious complications because of uncertainties over future states which the model must reflect. In that context, the existence of prices that bring supply and demand into balance is far from assured, and current theory is unsatisfying. However, strong results about the existence of equilibrium have recently been obtained by requiring utility functions to be concave instead of just quasi-concave while taking a new approach to "money". Concavity allows Lagrangian duality to generate underlying compactness in variational inequality representation of equilibrium. In combination with the "money" feature, it greatly simplifies the assumptions needed on the initial holdings of the agents.

SI spaces, maximal monotonicity and SN spaces

Stephen Simons

Department of Mathematics, University of California, Santa Barbara

Abstract

We introduce $SI$ and $SN$ spaces, which include Hilbert spaces, negative Hilbert spaces and spaces of the form $E\times E^*$, where $E$ is a nonzero real Banach space. We introduce $L$-positive subsets, which include monotone subsets of $E \times E^*$, and $BC$-functions, which include Fitzpatrick functions of monotone multifunctions. We show how convex analysis can be combined with $SI$ space theory to obtain and generalize various results on maximally monotone multifunctions on a reflexive Banach space, such as the significant direction of Rockafellar's surjectivity theorem and sufficient conditions for the sum of maximally monotone multifunctions to be maximally monotone. We will also discuss $L$-positive linear subspaces of SN spaces. This leads to generalizations of results of Bauschke-Borwein-Wang-Yao and Brezis-Browder. The second part of this talk uses the penalty function $\eta\|\cdot\|$ rather than the more usual $\frac{1}{2}\|\cdot\|^2.$

Generalized solutions for the sum of two maximally monotone operators

Walaa Moursi

Department of Mathematics, UBC Okanagan

Abstract

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this talk, we introduce a ''normal problem" associated with finding a zero of the sum of two maximally monotone operators. If the original problem admits solutions, then the normal problem returns this same set of solutions. The normal problem may yield solutions when the original problem does not admit any; furthermore, it has attractive variational and duality properties. Several examples illustrate our theory.

Slope and geometry in variational mathematics

Dmitriy Drusvyatskiy

Faculty of Mathematics, University of Waterloo

Abstract

Various notions of the "slope" of a real-valued function pervade optimization, and variational mathematics more broadly. In the semi-algebraic setting - an appealing model for concrete variational problems - the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. Highlights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the convergence of von Neumann's alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms.

[This is joint work with A. Daniilidis (Barcelona), A.D. Ioffe (Technion), M. Larsson (Lausanne), and A.S. Lewis (Cornell).]