Department of Computer Science & Engineering

&ldquo;IIT Jodhpur is committed to impart quality education to facilitate the creation of qualified, competent and responsible human resources to meet the emerging technological challenges of the world. It is working towards creating an equitable learning platform that is positive, accessible and effective in the face of the modern societal and scientific advancements.

Flexibility in curriculum, in all its academic programs, encompassing elements of teaching, research, creativity, innovation and entrepreneurship is the most unique feature of IIT Jodhpur. The Department of Science and Technology (DST), under the National Mission on Cyber Physical Systems, has identified IIT Jodhpur as a Technology &amp; Innovation Hub in the area of Augmented Reality and Virtual Reality. The institute has also been nominated as the coordinator for the Jodhpur City and Knowledge Cluster by the Office of Principal Scientific Adviser, Government of India.

The sprawling 852-acre IIT Jodhpur campus is a majestic tribute to the deep-rooted diverse cultural heritage of the region. The campus is an architectural masterpiece, which is slated to become an international exemplar of sustainability with net-zero energy, and advanced water and waste management strategies.&rdquo;

IIT Jodhpur

School of Artificial Intelligence and Data Science (AIDE)

All attempts to generalize the three-dimensional Lorenz model by selecting higher-order Fourier modes can be divided into three categories, namely: vertical, horizontal and vertical-horizontal mode truncations. The previous study showed that the first method allowed only construction of a nine-dimensional system when the selected modes were energy-conserving. The results presented in this paper demonstrate that a five-dimensional model is the lowest-order generalized Lorenz model that can be constructed by the second method and that its route to chaos is the same as that observed in the original Lorenz model. It is shown that the onset of chaos in both systems is determined by a number of modes that describe the vertical temperature difference in a convection roll. In addition, a simple rule that allows selecting modes that conserve energy for each method is derived. © 2006 Elsevier Ltd. All rights reserved.

World University root node

Chaos, Solitons and Fractals

Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations

The original 3D Lorenz model is one of the most celebrated nonlinear dynamical system. A number of attempts have been made to generalize this model to higher dimensions and investigate high-dimensional chaos. In this paper, the 3D Lorenz system is extended to higher dimensions by selecting higher-order horizontal and vertical modes in doubled Fourier expansions of a stream function and temperature variations. The selection of the modes is guided by the requirements that they must conserve energy in the dissipationless limit and lead to systems that have only bounded solutions. The obtained results showed that the lowest-order Lorentz model that satisfies these criteria is a 8D system, and that the onset of chaos and routes to chaos in this high-dimensional system are different than that observed in the original Lorenz model. Physical explanation of these results is given.

Proceedings of the 9th IASTED International Conference on Control and Applications, CA 2007

The onset of chaos and routes to chaos in high-dimensional Lorenz systems

To construct generalized Lorenz systems, higher-order modes in doubled Fourier expansions of a stream function and temperature variations must be considered. Selection of these modes is guided by the requirements that they conserve energy in the dissipationless limit and lead to systems that have bounded solutions. The previous study showed how to select the modes by using either vertical or horizontal mode truncations. In this paper, the most general method of horizontal and vertical mode truncations is presented and it is shown that the lowest-order generalized Lorenz system derived by this method is an eight dimensional system. An interesting result is that a route to chaos in this system is different than that observed in the original Lorenz model. Possible physical consequences of this result are discussed. © 2006 Elsevier Ltd. All rights reserved.

Generalized Lorenz models and their routes to chaos. III. Energy-conserving horizontal and vertical mode truncations

A two-dimensional and dissipative Rayleigh-Bénard convection can be approximated by the Lorenz model, which was originally derived by taking into account only three Fourier modes. Numerous attempts have been made to generalize this 3D model to higher dimensions and several different methods of selecting Fourier modes have been proposed. In this paper, generalized Lorenz models with dimensions ranging from four to nine are constructed by selecting vertical modes that conserve energy in the dissipationless limit and lead to systems that have bounded solutions. An interesting result is that the lowest-order generalized Lorenz model, which satisfies these criteria, is a 9D model and that its route to chaos is the same as that observed in the original 3D Lorenz model. The latter is in contradiction to some previous results that imply different routes to chaos in several generalized Lorenz systems. This discrepancy is explained by the fact that previously obtained generalized Lorenz models were derived by using different methods of selecting Fourier modes and, as a result, most of them did not obey the principle of conservation of energy in dissipationless limit. © 2006 Elsevier Ltd. All rights reserved.

Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations

Despite having a wide-spread applicability of evolutionary optimization procedures over the past few decades, EA researchers still face criticism about the theoretical optimality of obtained solutions. In this paper, we address this issue for problems for which gradients of objectives and constraints can be computed either exactly, or numerically or through subdifferentials. We suggest a systematic procedure of analyzing a representative set of Pareto-optimal solutions for their closeness to satisfying Karush-Kuhn-Tucker (KKT) points, which every Pareto-optimal solution must also satisfy. The procedure involves either a least-square solution or an optimum solution to a set of linear system of equations involving Lagrange multipliers. The procedure is applied to a number of differentiable and non-differentiable test problems and to a highly nonlinear engineering design problem. The results clearly show that EAs are capable of finding solutions close to theoretically optimal solutions in various problems. As a by-product, the error metric suggested in this paper can also be used as a termination condition for an EA application. Hopefully, this study will bring EAs and its research closer to classical optimization studies. © 2007 IEEE.

2007 IEEE Congress on Evolutionary Computation, CEC 2007

Finding trade-off solutions close to KKT points using evolutionary multi-objective optimization

The article is devoted to the study of the so-called optimistic version of bilevel programming in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation. Considering a basic optimistic model in bilevel programming, we reduce it to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model. Using advanced formulas for computing basic subgradients of value/marginal functions in variational analysis, we derive new necessary optimality conditions for bilevel programs reflecting significant phenomena that have never been observed earlier. In particular, our optimality conditions for bilevel programs do not depend on the partial derivatives with respect to parameters of the smooth objective function in the parametric lower-level problem. We present efficient implementations of our approach and results obtained for bilevel programs with differentiable, convex, linear, and Lipschitzian functions describing the initial data of the lower-level and upper-level problems.

Optimization

New necessary optimality conditions in optimistic bilevel programming

In this article we discuss the conditions required to guarantee the non-emptiness and the boundedness of certain subsets of the set of Lagrange multipliers for an inequality and equality constrained vector minimization problem.

Journal of Global Optimization

Bounded sets of KKT multipliers in vector optimization

Focus in the article is on necessary optimality conditions for bilevel programming problems. We first identify approaches that seem to be promising. Then, two such approaches are investigated. Using the nonexistence of a descent direction for the objective function within the tangent cone to the feasible set as necessary optimality condition we need efficient ways to describe this tangent cone. We describe possibilities for this in three different special cases. We conclude this article with the second approach applying Mordukhovich's coderivative to an appropriately reformulated bilevel programming problem. © 2006 Taylor & Francis.

Optimality conditions for bilevel programming problems

In this article we develop certain necessary optimality condition for bilevel programming problems with convex lower-level problem. The results are abstract in nature and depend on an important construction in nonsmooth analysis called the coderivative of a set-valued map. © 2006 Springer Science + Business Media, LLC.

Springer Optimization and Its Applications

Bilevel programming with convex lower level problems

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems. © Springer-Verlag 2006.