<p>To determine parameter values for models of reactions in the human body, like the glycolysis, good methods of parameter estimation are needed. Those models are often non-linear and estimation of the parameters can be very time consuming if it is possible at all. The goal of this work is to test different methods to improve the calculation speed of the parameter estimation of an example system. If the parameter estimation speed for the example system can be improved it is likely that the method could also be useful for systems similar to the example system.</p><p>One approach to improve the calculation speed is to construct a new cost function whose evaluation does not require any simulation of the system. Simulation free parameter estimation can be much quicker than using simulations to evaluate the cost function since the cost function is evaluated many times. Also a modication of the simulated annealing optimization method has been implemented and tested.</p><p>It turns out that some of the methods significantly reduced the time needed for the parameter estimations. However the quick methods have disadvantages in the form of reduced robustness. The most successful method was using a spline approximation together with a separation of the model into several submodels, and repeated use of the simulated annealing optimization algorithm to estimate the parameters.</p>

<p>In this thesis, we present algorithms for local and global minimization of some <i>Procrustes</i> type problems. Typically, these problems are about rotating and scaling a known set of data to fit another set with applications related to determination of rigid body movements, factor analysis and multidimensional scaling. The known sets of data are usually represented as matrices, and the rotation to be determined is commonly a matrix <i>Q</i> with orthonormal columns.</p><p>The algorithms presented use Newton and Gauss-Newton search directions with optimal step lengths, which in most cases result in a fast computation of a solution.</p><p>Some of these problems are known to have several minima, e.g., the weighted orthogonal Procrustes problem (WOPP). A study on the maximal amount of minima has been done for this problem. Theoretical results and empirical observations gives strong indications that there are not more than 2ⁿ minimizers, where <i>n</i> is the number of columns in <i>Q</i>. A global optimization method to compute all 2ⁿ minima is presented.</p><p>Also considered in this thesis is a cubically convergent iteration method for solving nonlinear equations. The iteration method presented uses second order information (derivatives) when computing a search direction. Normally this is a computational heavy task, but if the second order derivatives are constant, which is the case for quadratic equations, a performance gain can be obtained. This is confirmed by a small numerical study.</p><p>Finally, regularization of ill-posed nonlinear least squares problems is considered. The quite well known L-curve for linear least squares problems is put in context for nonlinear problems.</p>

In this thesis, we present algorithms for local and global minimization of some Procrustes type problems. Typically, these problems are about rotating and scaling a known set of data to fit another set with applications related to determination of rigid body movements, factor analysis and multidimensional scaling. The known sets of data are usually represented as matrices, and the rotation to be determined is commonly a matrix Q with orthonormal columns. The algorithms presented use Newton and Gauss-Newton search directions with optimal step lengths, which in most cases result in a fast computation of a solution. Some of these problems are known to have several minima, e.g., the weighted orthogonal Procrustes problem (WOPP). A study on the maximal amount of minima has been done for this problem. Theoretical results and empirical observations gives strong indications that there are not more than 2n minimizers, where n is the number of columns in Q. A global optimization method to compute all 2n minima is presented. Also considered in this thesis is a cubically convergent iteration method for solving nonlinear equations. The iteration method presented uses second order information (derivatives) when computing a search direction. Normally this is a computational heavy task, but if the second order derivatives are constant, which is the case for quadratic equations, a performance gain can be obtained. This is confirmed by a small numerical study. Finally, regularization of ill-posed nonlinear least squares problems is considered. The quite well known L-curve for linear least squares problems is put in context for nonlinear problems.

To determine parameter values for models of reactions in the human body, like the glycolysis, good methods of parameter estimation are needed. Those models are often non-linear and estimation of the parameters can be very time consuming if it is possible at all. The goal of this work is to test different methods to improve the calculation speed of the parameter estimation of an example system. If the parameter estimation speed for the example system can be improved it is likely that the method could also be useful for systems similar to the example system. One approach to improve the calculation speed is to construct a new cost function whose evaluation does not require any simulation of the system. Simulation free parameter estimation can be much quicker than using simulations to evaluate the cost function since the cost function is evaluated many times. Also a modication of the simulated annealing optimization method has been implemented and tested. It turns out that some of the methods significantly reduced the time needed for the parameter estimations. However the quick methods have disadvantages in the form of reduced robustness. The most successful method was using a spline approximation together with a separation of the model into several submodels, and repeated use of the simulated annealing optimization algorithm to estimate the parameters.

We introduce a very simple but efficient idea for branch & bound (B&B) algorithms in global optimization (GO). As input for our generic algorithm, we need an upper bound algorithm for the GO maximization problem and a branching rule. The latter reduces the problem into several smaller subproblems of the same type. The new B&B approach delivers one global optimizer or, if stopped before finished, improved upper and lower bounds for the problem. Its main difference to commonly used B&B techniques is its ability to approximate the problem from above and from below while traversing the problem tree. It needs no supplementary information about the system optimized and does not consume more time than classical B&B techniques. Experimental results with the maximum clique problem illustrate the benefit of this new method. (author's abstract) / Series: Working Papers on Information Systems, Information Business and Operations

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.

The process industries are characterized by the enormous use of natural resources such as raw materials, solvents, water, and utilities. Additionally, significant amounts of wastes are discharged from industrial facilities. As the world moves toward sustainable progress, that is, meeting the demand of the current generation without affecting or compromising the new generation, future process facilities must focus on resource conservation and pollution prevention. The purpose of this work is to introduce a new process integration methodology for the conservation and optimization of resources in the process industries. The work is also geared towards reducing waste discharge from the processing facilities. The optimal management of fresh resources and waste disposal requires the appropriate allocation, generation, and separation of streams and species. Material recycle/reuse/substitution, reaction alteration, and process modification are some of the main strategies employed to conserve resources in the process industries. The overall problem addressed in this dissertation can be stated as follows: Given is a process with a number of streams (sources) that are characterized by certain criteria (e.g., compositions of certain compounds, targeted properties) where these streams can be utilized in a number of process units (sinks) if they satisfy given constraints on flow rate, compositions, and/or properties. Additionally, interception devices may be used to adjust stream criteria. The objective is to develop targeting procedures and synthesis tools for the identification of minimum usage of fresh resources, minimum discharge of waste, and maximum integration of process resources. The devised methodology addresses four classes of problems: • Targeting techniques using direct recycle strategies • Recycle and interception procedures for single-component systems • Recycle and interception procedures for multi-component systems • Property integration for direct recycle strategies The framework provided by this dissertation couples traditional mass integration with groundbreaking property integration techniques to target, synthesize and optimize a plant for maximal conservation of resources. In particular, this work introduces new techniques such as material recycle pinch analysis, simultaneous recycle and interception networks, and property-based allocation. Additionally, graphical, algebraic, and optimization approaches are developed and validated with case studies in order to illustrate the applicability of the devised procedures.

Abstrakt: This report examines different control design methods, linear as well as nonlinear, for a suborbital reusable launch vehicle. An investigation of the natural vehicle behavior is made, after which a baseline linear controller is constructed to fulfill certain handling quality criteria. Thereafter the nonlinear cascade control methods block backstepping and nonlinear dynamic inversion via time scale separation are examined and used to construct two nonlinear controllers for the vehicle. Optimal controllers, in terms of three different criteria, are found using the genetic optimization algorithm differential evolution. The optimal controllers are compared, and it is found that nonlinear dynamic inversion via time scale separation performs better than block backstepping with respect to the cases investigated. The results suggest control design by global optimization is a viable and promising complement to classical methods.

<p>In this thesis, we devise a new finite branch-and-bound algorithm for disjoint bilinear programs. In these problems, there are two vectors of variables, $\b{x}$ and $\b{y}$, and two polytopes $P_{\b{x}}$ and $P_{\b{y}}$ such that $\b{x}$ (resp. $\b{y}$) is chosen from $P_{\b{x}}$ (resp. $P_{\b{y}}$) so that a bilinear objective function is minimized. By a bilinear objective, we mean that the objective becomes linear when either one of $\b{x}$ or $\b{y}$ is fixed. </p> <p>  This branch-and-bound scheme uses a relaxation that is derived using the reformulation-linearization technique (RLT). The RLT relaxation is constructed by taking products of constraints and linearizing the bilinear terms using introduced variables. The quality of this relaxation improves as higher order products and the corresponding monomial terms are linearized. Although it is known that, as RLT relaxations are constructed with increasingly higher order linearizations, the relaxation eventually converges to the true optimal value. However, no finite convergence properties are known. In contrast, we show that the first level of the RLT hierarchy suffices to convexify the problem when one of the polytopes is a simplex. Then, using this insight we devise a new class of relaxations by combining RLT with a variant of the double description procedure, where the latter lifts a polytope into a simplex in a finite number of steps.  This leads us to a finite hierarchy of relaxations that converges to the optimal value. Although this hierarchy is finite, from a computational perspective, we find that the relaxations grow rapidly in size. However, we utilize the insight to derive a simplicial branch-and-bound scheme, that expresses each polytope as a union of simplices allowing us to converge finitely to the optimal solution for the problem. We perform preliminary numerical experiments to show that this approach holds promise and competes favorably with state-of-the-art methods on larger instances.</p>

Nonsmoothness appears in various applications in chemical engineering, including multi-stream heat exchangers, nonsmooth flash calculation, process integration. In terms of numerical approaches, convex/concave relaxations of static and dynamic systems may also exhibit nonsmoothness. These relaxations are used in deterministic methods for global optimization. This thesis presents several new theoretical results for nonsmooth sensitivity analysis, with an emphasis on convex relaxations. Firstly, the "compass difference" and established ODE results by Pang and Stewart are used to describe a correct subgradient for a nonsmooth dynamic system with two parameters. This sensitivity information can be computed using standard ODE solvers. Next, this thesis also uses the compass difference to obtain a subgradient for the Tsoukalas-Mitsos convex relaxations of composite functions of two variables. Lastly, this thesis develops a new general subgradient result for Tsoukalas-Mitsos convex relaxations of composite functions. This result does not limit on the dimensions of input variables. It gives the whole subdifferential of Tsoukalas-Mitsos convex relaxations. Compare to Tsoukalas-Mitsos’ previous subdifferential results, it does not require additionally solving a dual optimization problem as well. The new subgradient results are extended to obtain directional derivatives for Tsoukalas-Mitsos convex relaxations. The new subgradient results and directional derivative results are computationally approachable: subgradients in this article can be calculated both by the vector forward AD mode and reverse AD mode. A proof-of-concept implementation in Matlab is discussed. / Thesis / Master of Applied Science (MASc)