Multilevel parametrization for aerodynamical optimization of 3D shapes

We introduce and discuss a combination of methods and options that aim at the aerodynamical optimization of a flow around an arbitrary aircraft shape. The flow is governed by the Euler equations, discretized by a mixed element-volume method on a fixed unstructured tetrahedrization. The shape parametrization relies on the skin of the above mesh through a hierarchical representation. Descent-type and one-shot algorithms are devised and adapted to the solution of a few model problems.     

Flow simulation and shape optimization for aircraft design

Within the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project.     

The performance of a Tau preconditioner on systems of ODEs

In this work, we investigate the new preconditioner of the Tau method, introduced by Ghoreishi and Mohammad Hosseini [13], on systems of ODEs. With a different and relatively straightforward view on the Tau formulation of systems of ODEs this preconditioner is reformulated and explained. An error analysis of the method is also addressed. Some numerical comparisons with the standard Chebyshev-Tau method and the preconditioned method of Cabos confirm the overall efficiency of this preconditioner for systems of ODEs as well.     

Optimum three-dimensional flight of a supersonic aircraft

Optimum three-dimensional flight of supersonic aircraft is considered, particularly the minimization of flight time, which is important in the climbing and turning performance of a supersonic fighter. General properties of optimum three-dimensional flight are presented, and the maximum principle is used. Optimal control is obtained by a geometrical method using the domain of manoeuvrability.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Necessary conditions for a domain optimization problem with various constraints

In this paper, a scheme for obtaining the necessary conditions in a wide class of domain optimization problems is given. In the case of a linear boundary-value problem of the Dirichlet type, necessary conditions are given.     

A new class of test functions for global optimization

In this paper we propose a new class of test functions for unconstrained global optimization problems. The class depends on some parameters through which the difficulty of the test problems can be controlled. As a basis for future comparison, we propose a selected set of these functions, with increasing difficulty, and some computational experiments with two simple global optimization algorithms.     

Scope of stationary multi-objective evolutionary optimization: a case study on a hydro-thermal power dispatch problem

Many engineering design and developmental activities finally resort to an optimization task which must be solved to get an efficient and often an intelligent solution. Due to various complexities involved with objective functions, constraints, and decision variables, optimization problems are often not adequately suitable to be solved using classical point-by-point methodologies. Evolutionary optimization procedures use a population of solutions and stochastic update operators in an iteration in a manner so as to constitute a flexible search procedure thereby demonstrating promise to such difficult and practical problem-solving tasks. In this paper, we illustrate the power of evolutionary optimization algorithms in handling different kinds of optimization tasks on a hydro-thermal power dispatch optimization problem: (i) dealing with non-linear, non-differentiable objectives and constraints, (ii) dealing with more than one objectives and constraints, (iii) dealing with uncertainties in decision variables and other problem parameters, and (iv) dealing with a large number (more than 1,000) variables. The results on the static power dispatch optimization problem are compared with that reported in an existing simulated annealing based optimization procedure on a 24-variable version of the problem and new solutions are found to dominate the solutions of the existing study. Importantly, solutions found by our approach are found to satisfy theoretical Kuhn–Tucker optimality conditions by using the subdifferentials to handle non-differentiable objectives. This systematic and detail study demonstrates that evolutionary optimization procedures are not only flexible and scalable to large-scale optimization problems, but are also potentially efficient in finding theoretical optimal solutions for difficult real-world optimization problems. Kalyanmoy Deb, Deva Raj Chair Professor. Currently a Finland Distinguished Professor, Department of Business Technology, Helsinki School of Economics, 00101 Helsinki, Finland.     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

Formulation of the problem of sonic boom by a maneuvering aerofoil as a one-parameter family of Cauchy problems

For the structure of a sonic boom produced by a simple aerofoil at a large distance from its source we take a physical model which consists of a leading shock (LS), a trailing shock (TS) and a one-parameter family of nonlinear wavefronts in between the two shocks. Then we develop a mathematical model and show that according to this model the LS is governed by a hyperbolic system of equations in conservation form and the system of equations governing the TS has a pair of complex eigenvalues. Similarly, we show that a nonlinear wavefront originating from a point on the front part of the aerofoil is governed by a hyperbolic system of conservation laws and that originating from a point on the rear part is governed by a system of conservation laws, which is elliptic. Consequently, we expect the geometry of the TS to be kink-free and topologically different from the geometry of the LS. In the last section we point out an evidence of kinks on the LS and kink-free TS from the numerical solution of the Euler’s equations by Inoue, Sakai and Nishida [5].     

Necessary optimality conditions for a class of nonsmooth vector optimization

The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of minimizing a vector function whose each component is the sum of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C of ℝ ⁿ , under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification. Supported by the National Natural Science Foundation of China (No. 70671064, No. 60673177), the Province Natural Science Foundation of Zhejiang (No.Y7080184) and the Education Department Foundation of Zhejiang Province (No. 20070306).     

Nonconvex optimization over a polytope using generalized capacity improvement

Capacity improvement involves the reduction of the size of the feasible region of an optimization problem without cutting-off the optimal solution point(s). Capacity improvement can be used in a branch-and-bound procedure to produce tighter relaxations to subproblems in the enumeration tree. Previous capacity improvement work has concentrated on tightening the simple lower and upper bounds on variables. In this paper, the capacity improvement procedure is generalized to apply toall constraints that form the feasible region. For the minimization of a separable concave function over a bounded polytope, the method of calculating the capacity improvement parameters is very straightforward. Computational results for fixed-charge and quadratic concave minimization problems demonstrate the effectiveness of this procedure.     

Importance of search-domain reduction in random optimization

The importance of incorporating systematic search-domain reduction into random optimization is illustrated. In the absence of domain reduction, even an enormous number of function evaluations does not ensure convergence sufficiently close to the optimum as was recently reported by Sarma. However, when the search domain is reduced systematically after every iteration as recommended by Luus and Jaakola, convergence is obtained in a relatively small number of function evaluations, even when the initial search region is large and the starting point is far from the optimum.     

Efficiency for a generalized form of vector optimization problems in complex space

A generalized form of vector optimization problems in complex space is considered, where both the real and the imaginary parts of the objective functions are taken into account. The efficient solutions are defined and characterized in terms of optimal solutions of related appropriate scalar optimization problems. These scalar problems are formulated by means of vectors in the dual of the domination cone. Under analyticity hypotheses about the functions, complex extensions to necessary and sufficient conditions for efficiency of Kuhn–Tucker type are established. Most of the corresponding results of previous studies (in both finite-dimensional complex and real spaces) can be recovered as particular cases.     

Formulation and optimization control of a class of networked evolutionary games with switched topologies

This paper studies the algebraic formulation and optimization control for a class of networked evolutionary games with switched topologies via the semi-tensor product method. First, an algebraic expression is formulated for the given networked evolutionary games, based on which, the behavior of corresponding evolutionary games is analyzed. Then, under some certain assumptions, the existence of fixed points for the given systems is proved and a free-type control sequence is designed to guarantee the best strategy profile reachable globally. Finally, an illustrative example is studied to support our new results.     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

Some characterizations of duality for DC optimization with composite functions

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively.     

An informational approach to the global optimization of expensive-to-evaluate functions

In many global optimization problems motivated by engineering applications, the number of function evaluations is severely limited by time or cost. To ensure that each evaluation contributes to the localization of good candidates for the role of global minimizer, a sequential choice of evaluation points is usually carried out. In particular, when Kriging is used to interpolate past evaluations, the uncertainty associated with the lack of information on the function can be expressed and used to compute a number of criteria accounting for the interest of an additional evaluation at any given point. This paper introduces minimizers entropy as a new Kriging-based criterion for the sequential choice of points at which the function should be evaluated. Based on stepwise uncertainty reduction, it accounts for the informational gain on the minimizer expected from a new evaluation. The criterion is approximated using conditional simulations of the Gaussian process model behind Kriging, and then inserted into an algorithm similar in spirit to the Efficient Global Optimization (EGO) algorithm. An empirical comparison is carried out between our criterion and expected improvement, one of the reference criteria in the literature. Experimental results indicate major evaluation savings over EGO. Finally, the method, which we call IAGO (for Informational Approach to Global Optimization), is extended to robust optimization problems, where both the factors to be tuned and the function evaluations are corrupted by noise.     

Optimality conditions for proper efficient solutions of vector set-valued optimization

Based on the concept of an epiderivative for a set-valued map introduced in J. Nanchang Univ. 25 (2001) 122-130, in this paper, we present a few necessary and sufficient conditions for a Henig efficient solution, a globally proper efficient solution, a positive properly efficient solution, an f-efficient solution and a strongly efficient solution, respectively, to a vector set-valued optimization problem with constraints.