Dynamic, convex, and robust optimization with Bayesian learning for response-guided dosing

Thesis (Ph.D.)--University of Washington, 2016-08

Existence of solutions to some equilibrium problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.

Stability for parametric implicit vector equilibrium problems

In this paper, we consider a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces where a mapping f and a set K are perturbed by parameters is an element of and lambda respectively. We establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution set mapping S : Lambda(1) x A(2) -> 2(X) for such parametric implicit vector equilibrium problems. (c) 2005 Elsevier Ltd. All rights reserved.

Transition in plane parallel shear flows heated internally

The stability of internally heated inclined plane parallel shear flows is examined numerically for the case of finite value of the Prandtl number, Pr. The transition in a vertical channel has already been studied for 0≤Pr≤100 with or without the application of an external pressure gradient, where the secondary flow takes the form of travelling waves (TWs) that are spanwise-independent (see works of Nagata and Generalis). In this work, in contrast to work already reported (J. Heat Trans. T. ASME 124 (2002) 635-642), we examine transition where the secondary flow takes the form of longitudinal rolls (LRs), which are independent of the steamwise direction, for Pr=7 and for a specific value of the angle of inclination of the fluid layer without the application of an external pressure gradient. We find possible bifurcation points of the secondary flow by performing a linear stability analysis that determines the neutral curve, where the basic flow, which can have two inflection points, loses stability. The linear stability of the secondary flow against three-dimensional perturbations is also examined numerically for the same value of the angle of inclination by employing Floquet theory. We identify possible bifurcation points for the tertiary flow and show that the bifurcation can be either monotone or oscillatory. © 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Perturbed Proximal Point Algorithm with Nonquadratic Kernel

Let H be a real Hilbert space and T be a maximal monotone operator on H. A well-known algorithm, developed by R. T. Rockafellar [16], for solving the problem (P) ”To find x ∈ H such that 0 ∈ T x” is the proximal point algorithm. Several generalizations have been considered by several authors: introduction of a perturbation, introduction of a variable metric in the perturbed algorithm, introduction of a pseudo-metric in place of the classical regularization, . . . We summarize some of these extensions by taking simultaneously into account a pseudo-metric as regularization and a perturbation in an inexact version of the algorithm.

Differential Equations in Abstract Cones

We extend the method of quasilinearization to differential equations in abstract normal cones. Under some assumptions, corresponding monotone iterations converge to the unique solution of our problem and this convergence is superlinear or semi–superlinear

The Perturbed Generalized Tikhonov's Algorithm

We work on the research of a zero of a maximal monotone operator on a real Hilbert space. Following the recent progress made in the context of the proximal point algorithm devoted to this problem, we introduce simultaneously a variable metric and a kind of relaxation in the perturbed Tikhonov’s algorithm studied by P. Tossings. So, we are led to work in the context of the variational convergence theory.

Generalized Variational Inequalities for Upper Hemicontinuous and Demi Operators with Applications to Fixed Point Theorems in Hilbert Spaces

∗ The final version of this paper was sent to the editor when the author was supported by an ARC Small Grant of Dr. E. Tarafdar.

Somme Ponctuelle D'operateurs Maximaux Monotones

∗ Cette recherche a été partiellement subventionnée, en ce qui concerne le premier et le dernier auteur, par la bourse OTAN CRG 960360 et pour le second auteur par l’Action Intégrée 95/0849 entre les universités de Marrakech, Rabat et Montpellier.

Stability of the Iteration Method for non Expansive Mappings

The general iteration method for nonexpansive mappings on a Banach space is considered. Under some assumption of fast enough convergence on the sequence of (“almost” nonexpansive) perturbed iteration mappings, if the basic method is τ−convergent for a suitable topology τ weaker than the norm topology, then the perturbed method is also τ−convergent. Application is presented to the gradient-prox method for monotone inclusions in Hilbert spaces.

The Boundary Descriptors of the n-dimensional Unit Cube Subset Partitioning

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

Generating More Boundary Elements of Subset Projections

Composition problem is considered for partition constrained vertex subsets of n dimensional unit cube E^n . Generating numerical characteristics of E^n subsets partitions is considered by means of the same characteristics in 1 − n dimensional unit cube, and construction of corresponding subsets is given for a special particular case. Using pairs of lower layer characteristic vectors for E^(1-n) more characteristic vectors for E^n are composed which are boundary from one side, and which take part in practical recognition of validness of a given candidate vector of partitions.

Semi-Smooth Newton Methods for the Time Optimal Control of Nonautonomous Ordinary Differential Equations

AMS Subj. Classification: 49J15, 49M15

Well-Posedness, Conditioning and Regularization of Minimization, Inclusion and Fixed-Point Problems

AMS subject classification: 65K10, 49M07, 90C25, 90C48.