In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an ffl-global optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include : (i) the general quadratic programming problem, (ii) quadratic programming problems with quadratic constraints, (iii) pooling and blending problems, and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach. Keywords : Global Optimization, Quadratic Programming, Quadratic Constraints, Polynomial functions, Pooling and Blending Problems. Author to whom...

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios or decomposing it into nodes corresponding to stages. In both cases the method has favorable convergence properties and a structure which makes it convenient for parallel computing environments. Keywords: Stochastic Programming, Decomposition, Augmented Lagrangian, Jacobi Method, Parallel Computation. iii iv On Augmented Lagrangian Decomposition Methods For Multistage Stochastic Programs Andrzej Ruszczy'nski 1 Introduction Multistage stochastic optimization problems belong to the most difficult problems of mathematical programming. Their size grows very quickly with the number of stages and with the number of events (scenarios) incorporated into the model. Although problems of this type occur frequently in applications (like,...

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

A decomposition method for large-scale convex optimization problems with block-angular structure and many linking constraints is analysed. The method is based on a separable approximation of the augmented Lagrangian function. Weak global convergence of the method is proved and speed of convergence analysed. It is shown that convergence properties of the method are heavily dependent on sparsity of the linking constraints. Application to large scale linear programming and stochastic programming is discussed. Key words: Large-Scale Optimization, Decomposition, Augmented Lagrangians. iii iv Augmented Lagrangian Decomposition For Sparse Convex Optimization Andrzej Ruszczy'nski 1. Introduction Rapid development of computing technology, emergence of parallel, massively parallel and distributed computing systems provides us with an increasing computing power but also creates a need for specialized approaches that can use it efficiently. The principal objective of this paper is to analys...

. This paper presents an overview of the deterministic global optimization approaches and their applications in the areas of Process Design, Control, and Computational Chemistry. The focus is on (i) decomposition-based primal dual methods, (ii) methods for generalized geometric programming problems, and (iii) global optimization methods for general nonlinear programming problems. The classes of mathematical problems that are addressed range from indefinite quadratic programming to concave programs, to quadratically constrained problems, to polynomials, to general twice continuously differentiable nonlinear optimization problems. For the majority of the presented methods nondistributed global optimization approaches are discussed with the exception of decomposition-based methods where a distributed global optimization approach is presented. 1. Background. A significant effort has been expended in the last five decades toward theoretical and algorithmic studies of applications that arise...

Prompted by advances in computer technology and the increasing confidence of decision makers in large-scale market models, practitioners of operations research are now tackling problems of increasing detail, complexity and size. This necessitates the development of new solution algorithms that exploit problem structure as well as the properties of the target hardware, in order to minimize turnaround time and maximize model utilization. Many models in planning and scheduling exhibit a block-angular structure, that can represent spatial or temporal partial decomposability: decision variables can be broken down to largely independent blocks, that correspond to first-level decisions satisfying a subset of the constraints, which may represent a time period, or a geographical region, or a commodity. The blocks interact via coupling constraints related to second-level coordination of block decisions, such as shared resource allocation restrictions. In this thesis we construct three efficient decomposition algorithms for such block-angular problems. These algorithms belong to the family of alternating directions methods, and can be thought of as block Gauss-Seidel iterative schemes for an augmented Lagrangian, that exploit the block structure. Alternatively, they can be thought of as Douglas--Rachford schemes for calculating a zero of the maximal monotone subgradient operator. Our algorithms are of the "fork--join" type, alternating a local and a global computation phase. In the local phase, decoupled optimization subproblems corresponding to blocks are solved. In the global phase, solution information is combined and a coordination problem is solved, the results of which are used in modifying the objective function of the subproblems. The algorithms are thus similar to price-d...

We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems. Keywords: parallel computing, alter...

Recent advances in global optimization for process synthesis, design and control are discussed. After a review of the chemical engineering contributions, we focus on the enclosure of all solutions of nonlinear constrained systems of equations. Important theoretical results are presented accompanied with computational studies on the enclosure of multiple steady states and all homogeneous azeotropes. 1 Introduction and Review A significant effort has been expended in the last four decades toward theoretical and algorithmic studies of applications that arise in Chemical Engineering Process Design, Process Synthesis, Process Control, as well as in Computational Chemistry and Molecular Biology. In the last decade we have experienced a dramatic growth of interest in Chemical Engineering for new methods of global optimization and their application to important engineering, as well as computational chemistry and molecular biology problems. Contributions from the chemical engineering communit...

Abstract. In order for primal-dual methods to be applicable to a constrained minimization problem, it is necessary that restrictive convexity conditions are satisfied. In this paper, we consider a procedure by means of which a nonconvex problem is convexified and transformed into one which can be solved with the aid of primal-dual methods. Under this transformation, separability of the type necessary for application of decomposition algorithms is preserved. This feature extends the range of applicability of such algorithms to nonconvex problems. Relations with multiplier methods are explored with the aid of a local version of the notion of a conjugate convex function. Key Words. Primal-dual methods, convexification procedures, decomposition methods, multiplier methods, local convex conjugate functions. 1.

: This paper presents an overview of the recent advances in deterministic global optimization approaches and their applications in the areas of Process Design and Control. The focus is on global optimization methods for (a) twice-differentiable constrained nonlinear optimization problems, (b) mixed-integer nonlinear optimization problems, and (c) locating all solutions of nonlinear systems of equations. Theoretical advances and computational studies on process design, batch design under uncertainty, phase equilibrium, location of azeotropes, stability margin, process synthesis, and parameter estimation problems are discussed. Keywords: Global Optimization; Twice Differentiable NLPs; Mixed-Integer Nonlinear Optimization; Locating All Solutions; ffBB approach, Design and Control 1. INTRODUCTION A significant effort has been expended in the last five decades toward theoretical and algorithmic studies of applications that arise in Process Design and Control. In the last decade we have expe...