On solving large systems of polynomial equations appearing in discrete differential geometry

The paper describes methods for solving very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations Q ₃=0 on an elementary cubic cell of the lattice ℤ³. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A “probing” technique, which replaces independent variables by random integers or zero, allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem. The text was submitted by the author in English.     

Gradient-based Pareto front approximation applied to turbomachinery shape optimization

Engineering with Computers - Multi-objective optimization has been rising in popularity, especially within an industrial environment, where several cost functions often need to be considered during...     

Approximating the nondominated front using the Pareto Archived Evolution Strategy

We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + lambda) and (mu + lambda) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks.     

Adaptive weighted-sum method for bi-objective optimization: Pareto front generation

This paper presents a new method that effectively determines a Pareto front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.     

Searching for knee regions of the Pareto front using mobile reference points

Evolutionary Algorithms (EAs) have been recognized to be well suited to approximate the Pareto front of Multi-objective Optimization Problems (MOPs). In reality, the Decision Maker (DM) is not interested in discovering the whole Pareto front rather than finding only the portion(s) of the front that matches at most his/her preferences. Recently, several studies have addressed the decision-making task to assist the DM in choosing the final alternative. Knee regions are potential parts of the Pareto front presenting the maximal trade-offs between objectives. Solutions residing in knee regions are characterized by the fact that a small improvement in either objective will cause a large deterioration in at least another one which makes moving in either direction not attractive. Thus, in the absence of explicit DM’s preferences, we suppose that knee regions represent the DM’s preferences themselves. Recently, few works were proposed to find knee regions. This paper represents a further study in this direction. Hence, we propose a new evolutionary method, denoted TKR-NSGA-II, to discover knee regions of the Pareto front. In this method, the population is guided gradually by means of a set of mobile reference points. Since the reference points are updated based on trade-off information, the population converges towards knee region centers which allows the construction of a neighborhood of solutions in each knee. The performance assessment of the proposed algorithm is done on two- and three-objective knee-based test problems. The obtained results show the ability of the algorithm to: (1) find the Pareto optimal knee regions, (2) control the extent (We mean by extent the breadth/spread of the obtained knee region.) of the obtained regions independently of the geometry of the front and (3) provide competitive and better results when compared to other recently proposed methods. Moreover, we propose an interactive version of TKR-NSGA-II which is useful when the DM has no a priori information about the number of existing knees in the Pareto optimal front.     

Pareto analysis of evolutionary and learning systems

This paper attempts to argue that most adaptive systems, such as evolutionary or learning systems, have inherently multiple objectives to deal with. Very often, there is no single solution that can optimize all the objectives. In this case, the concept of Pareto optimality is key to analyzing these systems. To support this argument, we first present an example that considers the robustness and evolvability trade-off in a redundant genetic representation for simulated evolution. It is well known that robustness is critical for biological evolution, since without a sufficient degree of mutational robustness, it is impossible for evolution to create new functionalities. On the other hand, the genetic representation should also provide the chance to find new phenotypes, i.e., the ability to innovate. This example shows quantitatively that a trade-off between robustness and innovation does exist in the studied redundant representation. Interesting results will also be given to show that new insights into learning problems can be gained when the concept of Pareto optimality is applied to machine learning. In the first example, a Pareto-based multi-objective approach is employed to alleviate catastrophic forgetting in neural network learning. We show that learning new information and memorizing learned knowledge are two conflicting objectives, and a major part of both information can be memorized when the multi-objective learning approach is adopted. In the second example, we demonstrate that a Pareto-based approach can address neural network regularizationmore elegantly. By analyzing the Pareto-optimal solutions, it is possible to identifying interesting solutions on the Pareto front.     

A procedure to prove statements in differential geometry

An algorithm for theorem proving in differential geometry based on the calculation of the differential dimension of differential quasi-algebraic sets is shown. In the case in which only ordinary differential equations are involved, an algorithm for such computation is presented. Different notions of validity for differential geometry statements are also compared.This paper was supported by Italian M.P.I. (40% 1985).     

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.     

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.     

A flexible cluster-oriented alternative clustering algorithm for choosing from the Pareto front of solutions

Supervised alternative clustering is the problem of finding a set of clusterings which are of high quality and different from a given negative clustering. The task is therefore a clear multi-objective optimization problem. Optimizing two conflicting objectives at the same time requires dealing with trade-offs. Most approaches in the literature optimize these objectives sequentially (one objective after another one) or indirectly (by some heuristic combination of the objectives). Solving a multi-objective optimization problem in these ways can result in solutions which are dominated, and not Pareto-optimal. We develop a direct algorithm, called COGNAC, which fully acknowledges the multiple objectives, optimizes them directly and simultaneously, and produces solutions approximating the Pareto front. COGNAC performs the recombination operator at the cluster level instead of at the object level, as in the traditional genetic algorithms. It can accept arbitrary clustering quality and dissimilarity objectives and provides solutions dominating those obtained by other state-of-the-art algorithms. Based on COGNAC, we propose another algorithm called SGAC for the sequential generation of alternative clusterings where each newly found alternative clustering is guaranteed to be different from all previous ones. The experimental results on widely used benchmarks demonstrate the advantages of our approach.

     

Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation

This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the -dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies. Presented as paper AIAA-2004-4322 at the 10th AIAA-ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, August 30–September 1, 2004     

On application of differential geometry to computational mechanics

Developments in the fields of computational science—the finite element method—and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13–15] or modeling mechanical features of engineering problems under consideration [4–7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry.

Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. ³. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or C^r differentiable atlases which are compatible.

In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas.

The properties of geodesic curves have been used by Yuan et al. [13–15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four- and eight-node elements in ². The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in ³.

Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems.

Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in ² for which geodesic coordinates are used as local coordinates.     

Matrix-free algorithm for the optimization of multidisciplinary systems

Multidisciplinary engineering systems are usually modeled by coupling software components that were developed for each discipline independently. The use of disparate solvers complicates the optimization of multidisciplinary systems and has been a long-standing motivation for optimization architectures that support modularity. The individual discipline feasible (IDF) formulation is particularly attractive in this respect. IDF achieves modularity by introducing optimization variables and constraints that effectively decouple the disciplinary solvers during each optimization iteration. Unfortunately, the number of variables and constraints can be significant, and the IDF constraint Jacobian required by most conventional optimization algorithms is prohibitively expensive to compute. Furthermore, limited-memory quasi-Newton approximations, commonly used for large-scale problems, exhibit linear convergence rates that can struggle with the large number of design variables introduced by the IDF formulation. In this work, we show that these challenges can be overcome using a reduced-space inexact-Newton-Krylov algorithm. The proposed algorithm avoids the need for the explicit constraint Jacobian and Hessian by using a Krylov iterative method to solve the Newton steps. The Krylov method requires matrix-vector products, which can be evaluated in a matrix-free manner using second-order adjoints. The Krylov method also needs to be preconditioned, and a key contribution of this work is a novel and effective preconditioner that is based on approximating a monolithic solution of the (linearized) multidisciplinary system. We demonstrate the efficacy of the algorithm by comparing it with the popular multidisciplinary feasible formulation on two test problems.     

Periodic value problems of differential systemson infinite-dimensional spaces and applications to differential geometry

Periodicity questions of differential systems on infinite-dimensional Hilbert spaces arestudied via a new methodology which is based on Fan-Knaster-Kuratowski-Mazurkiewicz theorem. We obtain in this way an alternative, to the classical fixed point theory, approach to the study of such type of problems with various applications on issues of mathematical analysis and differential geometry. Two examples of such applications are included.     

Obstacle recognition in front of vehicle based on geometry information and corrected laser intensity

Accurate identification and classification of obstacles in front of vehicle is an important part of intelligent vehicle safety driving. To solve the difficulties in distinguishing geometry similar obstacles and blocked obstacles, a real-time algorithm to accurately identify vehicles and pedestrians in a single frame was presented using laser intensity correction model and obstacle characteristic information. To complete the identification of obstacles, there are two classifications; the first classification according to the diagonal lengths of obstacles’ minimum enclosing rectangle, and then the second classification according to the mean and variance of intensity. As for different overlap criterion, result shows that the classification performance of our methods is better than other methods available.     

Absolute stability of positive systems with differential constraints

We consider discrete-time systems x(k + 1) = Ax(k) + f(x(k)) where the matrix A of the linear part is known and positive, the non-linearity f is unknown but belongs to a class for which A + f(x) is positive with spectral radius < 1 for all x Rⁿ. This, with the additional property that x - Ax - f (x) is proper, is sufficient for global stability of the system. The results are applied to the continuous system x. = Ax + B phialt(C^T x) by considering the translation operator along trajectories and studying the resulting discrete system.