Tapio Westerlund

Convex underestimation strategies for signomial functions

Different types of underestimation strategies are used in deterministic global optimization. In this paper, convexification and underestimation techniques applicable to problems containing signomial functions are studied. Especially, power transformation and exponential transformation (ET) will be considered in greater detail and some new theoretical results regarding the relation between the negative power transformation and the ET are given. The techniques are, furthermore, illustrated through examples and compared with other underestimating methods used in global optimization solvers such as αBB and BARON.

Some Transformation Techniques in Global Optimization

In this chapter some transformation techniques, useful in deterministic global optimization, are discussed. With the given techniques, a general class of nonconvex MINLP (mixed integer non-linear programming) problems can be solved to global optimality. The transformations can be applied to signomial functions and the feasible region of the original problem can be convexified and overestimated by the transformations. The global optimal solution of the original nonconvex problem can be found by solving a sequence of convexified MINLP sub-problems. In each such iteration a part of the infeasible region is cut off and the algorithm terminates when a solution point is sufficiently close to or within the feasible region of the original problem. The principles behind the algorithm are given in this chapter and numerical examples are used to illustrate how the global optimal solution is obtained with the algorithm.

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

On the modeling of mineral fiber formation

Abstract Some models for the investigation of the breaking mechanisms in the formation of mineral fibers with a “spinning machine” are presented in the paper. The temperature, position and internal tension of a single fiber during the formation phase (when the fiber is on the spinning disc) are calculated according to the effects of the inertial and of aerodynamic forces. Conclusions about the breaking mechanisms of the fiber may be drawn on the basis of the calculations for the tension and the temperature of the fiber in question.

On the uniqueness in pH calculations

Abstract Calculation of the pH value in a mixture of an arbitrary number of m -basic acids and l -acidic bases is considered. It is shown that there exists only one unique hydroxonium ion activity satisfying the nonlinear equation from which the pH value can be calculated. Furthermore some emphasis has been put on the numerical solution of the equation and an algorithm with global convergence is proposed.

Constrained linear quadratic gaussian control

Linear quadratic gaussian control with explicit input and state variance restrictions is discussed. Various topics in the design procedure are considered, for example the use of low-order or structurally constrained regulators. Numerical algorithms for solving the Constrained LQG control problems are given facilitating the application of the Constrained LQG design procedure. Two examples illustrating the design procedure are described. The examples are taken from a process for the production of plastic film and from a chemical reactor control system described in the literature.

PARALLEL SOLUTION OF DISJUNCTIVE MINLP PROBLEMS

In this paper, a system for parallel solution of large scale disjunctive optimization problems is presented. Disjunctive optimization problems occur in process engineering; both in production scheduling and in process synthesis. In general these problems are computationally very hard to solve. This system combines the demands of process engineering optimization problems with the potential CPU-power of the computers connected to the Internet. The system is capable of using an arbitrary number of computers connected to the Internet for distributed solution of rigorous industrial optimization problems.

Factorization of reaction systems applied to catalytic reactions

Abstract A procedure for factorization of reaction systems using an external and an internal set of species is introduced. The factorization method is applied to catalytic reactions where the external species represent bulk phase species and the internal species vacant sites and adsorbed intermediates on the catalyst surface. It is illustrated that several catalytic reaction schemes may be found describing the overall bulk phase reactions. The proposed factorization method gives a systematic procedure for discovering possible reaction mechanisms starting from a given set of chemical species or a given set of reactions.

Experiences from a Digital Quality Control System for Cement Kilns

Abstract High energy costs during the last decade has in most cement factories resulted in a change from the wet mehod of cement making. Also the use of coal as fuel instead of oil has become ever more popular. These changes do not only have had a positive effect on the process although the fuel economy has been improved. The dry method implies homogenization of a dry raw material, which technically most often result in a more inhomogenious raw meal than the slurry used in the wet method. Also the change of fuel from oil to coal has lead to a more fluctuating energy flow to the cement kiln, resulting in undesirable kiln behaviour and deteriorated clinker quality. These aspects have motivated the development of efficient control strategies for cement kilns, in order to get the maximum profit of the process changes. In this paper a digital quality control system which has succesfully been in on-line operation on two dry process cement kilns at the cement factory of the Lohja corporation in Virkkala, Finland, is presented. Each kiln have the annual capacity of 300.000 tons of clinker and the control system has been in on-line operation on the first kiln since 1980 and on the other one since april 1982.

EXPERIENCES FROM A DIGITAL QUALITY CONTROL SYSTEM FOR CEMENT KILNS

High energy costs during the last decade has in most cement ——– factories resulted in a change from the wet mehod of cement making. Also the use of coal as fuel instead of oil has become ever more popular. These changes do not only have had a positive effect on the process although the fuel economy has been improved. The dry method implies homogenization of a dry raw material, which technically most often result in a more inhomogenious raw meal than the slurry used in the wet method. Also the change of fuel from oil to coal has lead to a more fluctuating energy flow to the cement kiln, resulting in undesirable kiln behaviour and deteriorated clinker quality. These aspects have motivated the development of efficient control strategies for cement kilns, in order to get the maximum profit of the process changes. In this paper a digital quality control systemwhich has succesfully been in on-line operation on two dry process cement kilns at the cement factory of the Lohja corporation in Virkkala, Finland, is presented. Each kiln have the annual capacity of 300.000tons of clinker and the control system has been in on-line operation on the first kiln since 1980 and on the other one since april 1982.