Department of Computer Science & Engineering

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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)

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations - one involving Lagrange multipliers and other involving decision variables - to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study. © 2012 Springer Science+Business Media New York.

World University root node

Journal of Global Optimization

Solving dual problems using a coevolutionary optimization algorithm

Postinhibitory rebound (PIR) is believed to play an important role in the genesis and maintenance of biological rhythms. While it has been demonstrated during several in vitro studies, in vivo evidence for PIR remains scarce. Here, we report that PIR can be observed in the dorsomedial entorhinal cortex of anesthetizedrats, mostly between putatively connected GAB Aergic interneurons, and that it is more prevalent during the theta (4 - 6 Hz) oscillation state than the slow (0.5-2 Hz) oscillation state. Functional inhibition was also found to be brain state and postsynaptic cell type dependent but that alone could not explain this brain state dependence of PIR. A theoretical analysis, using two Fitzhugh-Nagumo neurons coupled to an external periodic drive, predicted that the modulation of a faster spiking rate by the slower periodic drive could account for the brain state dependence of PIR. Model predictions were verified experimentally. We conclude that PIR is cell type and brain state dependent and propose that this could impact network synchrony and rhythmogenesis. © 2012 the authors.

Fulltext

Journal of Neuroscience

Brain State Dependent Postinhibitory Rebound in Entorhinal Cortex Interneurons

Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is "No" even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater's constraint qualification. We also show by examples that this correspondence can fail if the Slater's constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples. © 2010 Springer and Mathematical Programming Society.

Mathematical Programming

Is bilevel programming a special case of a mathematical program with complementarity constraints?

Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network's phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network's phase description. © 2011 American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Phase description of spiking neuron networks with global electric and synaptic coupling

In this article we introduce the notion of approximate Karush-Kuhn- Tucker (KKT) points for smooth, convex and nonsmooth, nonconvex vector optimization problems. We study a kind of stability of these points and KKT points of vector optimization problems. In the convex case we also introduce and study the notion of modified approximate KKT points motivated by Ekeland's variational principle. We prove stability properties of these points for several optimization problems. © 2011 Taylor & Francis.

Optimization

Stability properties of KKT points in vector optimization

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts. © 2011 Copyright Taylor and Francis Group, LLC.

A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

Gap functions have proved to be efficient tools to study single-valued variational inequalities. This approach allows us to reformulate the problem into an optimization problem. New notions of gap functions are defined for set-valued variational inequalities. We prove finiteness and error bounds properties, i.e. upper estimates for the distance to the solution set of the variational inequality. © 2011 Springer Science+Business Media, LLC.

Journal of Optimization Theory and Applications

On Gap Functions for Multivalued Stampacchia Variational Inequalities

In this paper we introduce a scalar gap function for a Stampacchia type vector variational inequality problem. We show how properties of the gap function can be improved by regularizing it in the sense of Fukushima. We study in detail the properties of this regularized gap function. An error bound for the vector variational inequality is obtained in terms of the regularized gap function. We also introduce the notion of D-gap function for vector variational inequalities. We study its properties and also present an error bound in terms of the D-gap functions. © 2010 Yokohama Publishers.

Pacific Journal of Optimization

Regularized gap functions and error bounds for vector variational inequalities

We present a new family of proximal point methods for solving monotone variational inequalities. Our algorithm has a relative error tolerance criterion in solving the proximal subproblems. Our convergence analysis covers a wide family of regularization functions, including double regularizations recently introduced by Silva, Eckstein, and Humes, Jr.[SIAM J. Optim., 12 (2001), pp. 238-261] and the Bregman distance induced by h(x) = ∑i=1 n xi log xi We do not use in our analysis the assumption of paramonotonicity, which is standard in proving convergence of Bregman-based proximal methods. © by SIAM.

SIAM Journal on Optimization

Inexact proximal point methods for variational inequality problems

The problem to find a best solution within the set of optimal solutions of a convex optimization problem is modeled as a bilevel programming problem. It is shown that regularity conditions like Slater’s constraint qualification are never satisfied for this problem. If the lower-level problem is replaced with its (necessary and sufficient) optimality conditions, it is possible to derive a necessary optimality condition for the resulting problem. An example is used to show that this condition in not sufficient even if the initial problem is a convex one. If the lower-level problem is replaced using its optimal value, it is possible to obtain an optimality condition that is both necessary and sufficient in the convex case. © Springer Science+Business Media, LLC 2010.