Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints

Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H\"{o}lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique.Comment: 29 pages, 5 figure

A stochastic smoothing method for nonsmooth global optimization

The paper presents the results of testing the stochastic smoothing method for global optimization of a multiextremal function in a convex feasible subset of the Euclidean space. Preliminarily, the objective function is extended outside the admissible region so that its global minimum does not change, and it becomes coercive.Проблема глобальної оптимізації неопуклих негладких функцій з обмеженнями є актуальною для багатьох інженерних застосувань, зокрема, для навчання неопуклих негладких нейронних мереж. У роботі представлені результати тестування методу згладжування багато екстремальної цільової функції для знаходження її глобального мінімуму в деякої опуклій допустимій області евклідового простору. Попередньо цільова функція довизначається поза опуклої допустимої області так, щоб не змінити її глобального мінімуму, та зробити її коерцитивною.Проблема глобальной оптимизации невыпуклых негладких функций при ограничениях актуальна для многих инженерных приложений, в частности, для обучения невыпуклых негладких нейронных сетей. В работе представлены результаты тестирования метода сглаживания многоэкстремальной целевой функции для нахождения ее глобального минимума в некоторой выпуклой допустимой области евклидового пространства. Предварительно целевая функция доопределяется вне допустимой области так, чтобы не изменить ее глобальный минимум, и сделать ее коэрцитивной

Nondifferentiable Optimization: Motivations and Applications

IIASA has been involved in research on nondifferentiable optimization since 1976. The Institute's research in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimization has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition and to review developments in this field IIASA held this Workshop in Sopron (Hungary) in September 1984. This volume contains selected papers presented at the Workshop. It is divided into four sections dealing with the following topics: (I) Concepts in Nonsmooth Analysis; (II) Multicriteria Optimization and Control Theory; (III) Algorithms and Optimization Methods; (IV) Stochastic Programming and Applications

Productivity enhancement through process integration

A hierarchical procedure is developed to determine maximum overall yield of a process and optimize process changes to achieve such a yield. First, a targeting procedure is developed to identify an upper bound of the overall yield ahead of detailed design. Several mass integration strategies are proposed to attain maximum yield. These strategies include rerouting of raw materials, optimization of reaction yield, rerouting of product from undesirable outlets to desirable outlets, and recycling of unreacted raw materials. Path equations are tailored to provide the appropriate level of detail for modeling process performance as a function of the optimization variables pertaining to design and operating variables. Interval analysis is used as an inclusion technique that provides rigorous bounds regardless of the process nonlinearities and without enumeration. Then, a new approach for identification of cost-effective implementation of maximum attainable targets for yield is presented. In this approach, a mathematical program was developed to identify the maximum feasible yield using a combination of iterative additions of constraints and problem reformulation. Next, cost objectives were employed to identify a cost-effective solution with the details of design and operating variables. Constraint convexification was used to improve the quality of the solution towards globability. A trade-off procedure between the saving and expenses for yield maximization problem is presented. The proposed procedure is systematic, rigorous, and computationally efficient. A case study was solved to demonstrate the applicability and usefulness of the developed procedure

Global optimisation in process design

This thesis concerns the development of rigorous global optimisation techniques and their application to process engineering problems. Many Process Engineering optimisation problems are nonlinear. Local optimisation approaches may not provide global solutions to these problems if they are nonconvex. The global optimisation approach utilised in this work is based on interval branch and bound algorithms. The interval global optimisation approach is extended to take advantage of information about the structure of the problem and facilitate efficient solution of constrained NLPs using interval analysis. This is achieved by reformulating the interval lower bounding procedure as a convex programming problem which allows inclusion of convex constraints in the lower bounding problem. The approach is applied to a number of standard constrained test problems indicating that this algorithm retains the wide applicability of the interval methods while allowing efficient solution of constrained problems. A new approach to the construction of modular flowsheets is developed. This approach allows construction of flowsheets from linked unit models which enable the application of a number of global optimisation algorithms. The modular flowsheets are constructed with 'generic' unit operations which provide interval bounds, linear bounds, derivatives and derivative bounds using extended numerical types. The genericity means that new 'extended types' can be devised and used without rewriting the unit operations models. The new interval global optimisation algorithm is applied to the generic modular flowsheet. Using interval analysis and automatic differentiation as the arithmetic types, lower bounding linear programs are constructed and used in a branch and bound framework to globally optimise the modular flowsheet

Derivative free algorithms for nonsmooth and global optimization with application in cluster analysis

This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.Doctor of Philosoph