The aim of this work is to propose viscosity-like methods for finding a specific common fixed point of a finite family T={ T_{i} }_{i=1}^{N} of nonexpansive self-mappings of a closed convex subset C of a Hilbert space H.We propose two schemes: one implicit and the other explicit.The implicit scheme determines a set {x_{t} : 0 < t < 1} through the fixed point equation x_{t}= tf (x_{t} ) + (1− t)Tx_{t}, where f : C¡÷C is a contraction.The explicit scheme is the discretization of the implicit scheme and de defines a sequence {x_{n} } by the recursion x_{n+1}=£\\_{n}f(x_{n}) +(1−£\\_{n})Tx_{n} for n ≥ 0, where {£\\_{n} }⊂ (0,1) It has been shown in the literature that both implicit and explicit schemes converge in norm to a fixed point of T (with additional conditions imposed on the sequence {£\ _{n} } in the explicit scheme).We will extend both schemes to the case of a finite family of nonexpansive mappings. Our proposed schemes converge in norm to a common fixed point of the family which in addition solves a variational inequality.

The pooling problem is an important optimization problem that is encountered in operation and scheduling of important industrial processes within petroleum refineries. The key objective of pooling is to mix various intermediate products to achieve desired properties and quantities of products. First, intermediate streams from various processing units are mixed and stored in intermediate tanks referred to as pools. The stored streams in pools are subsequently allowed to mix to meet varying market demands. While these pools enhance the operational flexibility of the process, they complicate the decisionmaking process needed for optimization. The problem to find the least costly mixing recipe from intermediate streams to pools and then from pools to sale products is referred to as the pooling problem. The research objective is to contribute an approach to solve this problem. The pooling problem can be formulated as an optimization program whose objective is to minimize cost or maximize profit while determining the optimal allocation of intermediate streams to pools and the blending of pools to final products. Because of the presence of bilinear terms, the resulting formulation is nonconvex which makes it very difficult to attain the global solution. Consequently, there is a need to develop computationally-efficient and easy-to-implement global-optimization techniques to solve the pooling problem. In this work, a new approach is introduced for the global optimization of pooling problems. The approach is based on three concepts: linearization by discretizing nonlinear variables, pre-processing using implicit enumeration of the discretization to form a convex-hull which limits the size of the search space, and application of integer cuts to ensure compatibility between the original problem and the discretized formulation. The continuous quality variables contributing to bilinear terms are first discretized. The discretized problem is a mixed integer linear program (MILP) and can be globally solved in a computationally effective manner using branch and bound method. The merits of the proposed approach are illustrated by solving test case studies from literature and comparison with published results.

Given a connected oriented graph D, we say that a subset S of V(D) is convex in D if, for every pair of vertices x, y in S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity number con (D) of a nontrivial connected oriented graph D is the maximum cardinality of a proper convex set of D. Let S_{C}(K_{n})={con(D)|D is an orientation of K_{n}} and S_{SC}(K_{n})={con(D)|D is a strong orientation of K_{n}}. We show that S_{C}(K_{3})={1,2} and S_{C}(K_{n})={1,3,4,...,n-1} if n >= 4. We also have that S_{SC}(K_{3})={1} and S_{SC}(K_{n})={1,3,4,...,n-2} if n >= 4 . We also show that every triple n, m, k of integers with n >= 5, 3 <= k <= n-2, and n+1 <= m <= n(n-1)/2, there exists a strong connected oriented graph D of order n with |E(D)|=m and con (D)=k.

A mixture experiment is an experiments in which the q-ingredients are nonnegative and subject to the simplex restriction on the (q-1)-dimentional probability simplex. In this work , we investigate the robust A-optimal designs for mixture experiments with uncertainty on the linear, quadratic models considered by Scheffe' (1958). In Chan (2000), a review on the optimal designs including A-optimal designs are presented for each of the Scheffe's linear and quadratic models. We will use these results to find the robust A-optimal design for the linear and quadratic models under some robust A-criteria. It is shown with the two types of robust A-criteria defined here, there exists a convex combination of the individual A-optimal designs for linear and quadratic models respectively to be robust A-optimal. In the end, we compare efficiencies of these optimal designs with respect to different A-criteria.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 83-89). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 83-89). Also available on the Internet.

Thesis--Auburn University, 2009. / Abstract. Includes bibliographic references (p.70-72).

Thesis--M. Phil., University of Hong Kong, 1983.

This thesis is focused on the limits of performance of large-scale convex optimization algorithms. Classical theory of oracle complexity, first proposed by Nemirovski and Yudin in 1983, successfully established the worst-case behavior of methods based on local oracles (a generalization of first-order oracle for smooth functions) for nonsmooth convex minimization, both in the large-scale and low-scale regimes; and the complexity of approximately solving linear systems of equations (equivalent to convex quadratic minimization) over Euclidean balls, under a matrix-vector multiplication oracle. Our work extends the applicability of lower bounds in two directions: Worst-Case Complexity of Large-Scale Smooth Convex Optimization: We generalize lower bounds on the complexity of first-order methods for convex optimization, considering classes of convex functions with Hölder continuous gradients. Our technique relies on the existence of a smoothing kernel, which defines a smooth approximation for any convex function via infimal convolution. As a consequence, we derive lower bounds for \ell_p/\ell_q-setups, where 1\leq p,q\leq \infty, and extend to its matrix analogue: Smooth convex minimization (with respect to the Schatten q-norm) over matrices with bounded Schatten p-norm. The major consequences of this result are the near-optimality of the Conditional Gradient method over box-type domains (p=q=\infty), and the near-optimality of Nesterov's accelerated method over the cross-polytope (p=q=1). Distributional Complexity of Nonsmooth Convex Optimization: In this work, we prove average-case lower bounds for the complexity of nonsmooth convex ptimization. We introduce an information-theoretic method to analyze the complexity of oracle-based algorithms solving a random instance, based on the reconstruction principle. Our technique shows that all known lower bounds for nonsmooth convex optimization can be derived by an emulation procedure from a common String-Guessing Problem, which is combinatorial in nature. The derived average-case lower bounds extend to hold with high probability, and for algorithms with bounded probability error, via Fano's inequality. Finally, from the proposed technique we establish the equivalence (up to constant factors) of distributional, randomized, and worst-case complexity for black-box convex optimization. In particular, there is no gain from randomization in this setup.

published_or_final_version / Mathematics / Master / Master of Philosophy