A coarse solution of generalized semi-infinite optimization problems via robust analysis of marginal functions and global optimization

Die Arbeit beschäftigt sich überwiegend mit theoretischen Untersuchungen zur Bestimmung grober Startlösungen für verallgemeinerte semi-infinite Optimierungsaufgaben (GSIP) mit Methoden der globalen Optimierung. GSIP Probleme besitzen im Gegensatz zu den gewöhnlichen semi-infiniten Optimierungsaufgaben (SIP) die Eigenschaft, dass die Indexmenge, die die Restriktionen beschreibt, natürlich überabzählbar ist, wie bei (SIP) aber darüber hinaus von den Problemvariablen abhängig ist, d.h. die Indexmenge ist eine Punkt-Menge Abbildung. Solche Probleme sind von sehr komplexer Struktur, gleichzeitig gibt es große Klassen von naturwissenschaftlich - technischen, ökonomischen Problemen, die in (GSIP) modelliert werden können. Im allgemeinem ist die zulässige Menge von einem (GSIP) weder abgeschlossen noch zusammenhängend. Die Abgeschlossenheit von der zulässigen Menge ist gesichert durch die Unterhalbstetigkeit der Index-Abbildung. Viele Autoren machen diese Voraussetzung, um numerische Verfahren für (GSIP) herzuleiten. Diese Arbeit versucht erstmals, ohne Unterhalbstetigkeit der Index-Abbildung auszukommen. Unter diese schwächeren Voraussetzungen kann die zulässige Menge nicht abgeschlossen sein und (GSIP) kann auch keine Lösung besitzen. Trotzdem kann man eine verallgemeinerte Minimalstelle oder eine Minimalfolge für (GSIP) bestimmen. Für diese Zwecke werden zwei numerische Zugänge vorgeschlagen. Im ersten Zugang wird der zulässige Bereich des (GSIP) durch eine (gewöhnliche) parametrische semi- infinite Approximationsaufgabe beschrieben. Die Marginalfunktion der parametrischen Aufgabe ist eine exakte Straffunktion des zulässigen Bereiches des (GSIP). Im zweiten Zugang werden zwei Straffunktionen vorgestellt. Eine verwendet die semi-infinite Restriktion direkt als einen "Max"-Straffterm und die zweite entsteht durch das "lower level Problem" des (GSIP). In beiden Zugänge müssen wir uns mit unstetigen Optimierungsaufgaben beschäftigen. Es wird gezeigt, dass die entstehende Straffunktionen oberrobust (i.A. nicht stetig) sind und damit auch hier stochastische globale Optimierungsmethoden prinzipiell anwendbar sind. Der Hauptbeitrag dieser Arbeit ist die Untersuchung von Robustheiteigenschaften von Marginalfunktionen und Punkt-Menkg-Abbildung mit bestimmte Strukturen. Dieser kann auch als eine Erweiterung der Theorie der Robusten Analysis von Chew & Zheng betrachtet werden. Gleichzeitig wird gezeigt, dass die für halbstetigen Abbildungen und Funktionen bekannten Aussagen bis auf wenige Ausnahmen in Bezug auf das Robustheitskonzept übertragen werden können. Am Ende zeigen einige numerische Beispiele, dass die vorgeschlagenen Zugänge prinzipiell brauchbar sind.The aim of this work is to determine a coarse approximation to the optimal solution of a class of generalized semi-infinite optimization problems (GSIP) through a global optimization method by using fairly discontinuous penalty functions. Where the fairness of the discontinuities is characterized by the notions of robust analysis and standard measure theory. Generalized semi-infinite optimization problems have an infinite number of constraints, where the usually infinite index set of the constraints varies with respect to the problem variable; i.e. we have a set-valued map as and index set, in contrast to standard semi-infinite optimization (SIP) problems. These problems have very complex problem structures, at the same time, there are several classes of scientific, engineering, econimic, etc., problems which could be modelled in terms of (GSIP)s. Under general assumptions, the feasible set of a (GSIP) might not be closed nor connected. In fact, the feasible set is a closed set if the index map is lower semi-continuous. Several authors assume the lower semi-continuity of the index map for the derivation of numerical algorithms for (GSIP). However, in this work no exclusive assumption has been made to preserve the above nicer structures. Thus, the feasible set may not be closed and (GSIP) may not have a solution. However, one may be interested to determine a generalized minimizer or a minimizing sequence of GSIP. For this purpose, two penalty approaches have been proposed. In the first approach (mainly conceptual), there is defined a discontinuous penalty function based on the marginal function of a certain auxiliary parametric semi-infinite optimization problem (PSIP). In the second approach (based on discretization), we define two penalty functions: one based on the marginal function of the lower level problem and, a second, based on the feasible set of (GSIP). The relationships of these penalty problems with the (GSIP) have been investigated through minimizing sequences. In the two penalty approaches we need to deal with discontinuous optimization problems. The numerical treatment of these discontinuous optimization problems can be done by using the Integral Global Optimization Method (IGOM); in particular, through the software routine called BARLO (of Hichert). However, to use BARLO or IGOM we need to verify certain robustness properties of the objective functions of the penalty problems. Hence, one major contribution of this work is a study of robustness properties of marginal value functions and set-valued maps with given structures - extending the theory of robust analysis of Chew and Zheng. At the same time, an effort has been made to find out corresponding robustness results to some standard continuity notions of functions and set-valued maps. To show the viability of the proposed approach, numerical experiments are made using the penalty-discretization approach

Chance Constrained Optimal Power Flow Using the Inner-Outer Approximation Approach

In recent years, there has been a huge trend to penetrate renewable energy sources into energy networks. However, these sources introduce uncertain power generation depending on environmental conditions. Therefore, finding 'optimal' and 'feasible' operation strategies is still a big challenge for network operators and thus, an appropriate optimization approach is of utmost importance. In this paper, we formulate the optimal power flow (OPF) with uncertainties as a chance constrained optimization problem. Since uncertainties in the network are usually 'non-Gaussian' distributed random variables, the chance constraints cannot be directly converted to deterministic constraints. Therefore, in this paper we use the recently-developed approach of inner-outer approximation to approximately solve the chance constrained OPF. The effectiveness of the approach is shown using DC OPF incorporating uncertain non-Gaussian distributed wind power

Structural Properties and Convergence Approach for Chance-Constrained Optimization of Boundary-Value Elliptic Partial Differential Equation Systems

This work studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). The boundary conditions are random input functions deliberated from the boundary of the partial differential equation (PDE) system and in the infinite-dimensional reflexive and separable Banach space. The structural properties of the chance constraints studied in this paper are continuity, closedness, compactness, convexity, and smoothness of probabilistic uniform or pointwise state constrained functions and their parametric approximations. These are open issues even in the finite-dimensional Banach space. Thus, it needs finite-dimensional and smooth parametric approximation representations. We propose a convex approximation approach to nonconvex CCPDE problems. When the approximation parameter goes to zero from the right, the solutions of the relaxation and compression approximations converge asymptotically to the optimal solution of the original CCPDE. Due to the convexity of the problem, a global solution exists for the proposed approximations. Numerical results are provided to demonstrate the plausibility and applicability of the proposed approach

Modified multiple shooting combined with collocation method in JModelica.org with symbolic calculations

This paper presents an efficient and a novel implementation of a combined multiple shooting and collocation (CMSC) algorithm for the solution of nonlinear optimal control problems. The implemented algorithm is a modification of the approach proposed in [17; 18]. The new implementation is done under the JModelica.org framework along with CasADi and Ipopt. The framework uses a symbolic pre-calculation of functions and derivatives. Besides the integration of various components of JModelica.org; Ipopt; and CasADi; the implementation facilitates simpler modeling of optimal control problems along with a choice of options for various linear algebra algorithms. The paper gives a description of the algorithm and elaborates the components of the framework. Numerical experimentations show that the new implementation is efficient in comparison with the published results of other authors

Corruption dynamics: a mathematical model and analysis

This study proposes and analyzes a deterministic mathematical model to describe the dynamics of corruption transmission. We began by proving that the solution to the model is bounded and positive. The next-generation matrix approach is used to compute the basic reproduction number (R0) in relation to corruption-free equilibrium. The Jacobian and Lyapunov functions are used to show that corruption-free equilibrium is asymptotically stable in both locally and globally when R0<1, and otherwise, an endemic corruption equilibrium develops. Furthermore, the sensitivity of the model's parameters was investigated. The findings demonstrate that religious precepts govern public education. The two sectors most susceptible to corruption control are education and corrections. The study recommends investing more in the provision of public education to citizens by creating awareness among all and including it in the education curriculum and religious leaders to teach their followers seriously about the impact of corruption as well as the use of jail as punishment. The numerical simulation results agreed with the analytical results

On robustness of set-valued maps and marginal value functions

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7,26], and further developed by Shi, Zheng, Zhuang [18,19,20], Phú, Hoffmann and Hichert [8,9,10,17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also strongly convinced that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required

A Toolchain for Solving Dynamic Optimization Problems Using Symbolic and Parallel Computing

Abstract Significant progresses in developing approaches to dynamic optimization have been made. However, its practical implementation poses a difficult task and its realtime application such as in nonlinear model predictive control (NMPC) remains challenging. A toolchain is developed in this work to relieve the implementation burden and, meanwhile, to speed up the computations for solving the dynamic optimization problem. To achieve these targets, symbolic computing is utilized for calculating the first and second order sensitivities on the one hand and parallel computing is used for separately accomplishing the computations for the individual time intervals on the other hand. Two optimal control problems are solved to demonstrate the efficiency of the developed toolchain which solves one of the problems with approximately 25,000 variables within a reasonable CPU time