On Higher-order Duality in Nondifferentiable Minimax Fractional Programming

In this paper, we consider a nondifferentiable minimax fractional programming problem with continuously differentiable functions and formulated two types of higher-order dual models for such optimization problem.Weak, strong and strict converse duality theorems are derived under higherorder generalized invexity

On Minimax Fractional Optimality Conditions with Invexity

AbstractUnder different forms of invexity conditions, sufficient Kuhn–Tucker conditions and three dual models are presented for the minimax fractional programming

A Modified Levenberg-Marquardt Method for the Bidirectional Relay Channel

This paper presents an optimization approach for a system consisting of multiple bidirectional links over a two-way amplify-and-forward relay. It is desired to improve the fairness of the system. All user pairs exchange information over one relay station with multiple antennas. Due to the joint transmission to all users, the users are subject to mutual interference. A mitigation of the interference can be achieved by max-min fair precoding optimization where the relay is subject to a sum power constraint. The resulting optimization problem is non-convex. This paper proposes a novel iterative and low complexity approach based on a modified Levenberg-Marquardt method to find near optimal solutions. The presented method finds solutions close to the standard convex-solver based relaxation approach.Comment: submitted to IEEE Transactions on Vehicular Technology We corrected small mistakes in the proof of Lemma 2 and Proposition

Bounding extreme events in nonlinear dynamics using convex optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.Comment: Revised according to comments by reviewers. Added references and rearranged introduction, conclusions, and proofs. 38 pages, 7 figures, 4 tables, 4 appendices, 87 reference

On second order duality for nondifferentiable minimax fractional programming problems involving type-I functions

We introduce second order $(C,\alpha ,\rho ,d)$ type-I functions and formulate a second order dual model for a nondifferentiable minimax fractional programming problem. The usual duality relations are established under second order $(F,\alpha ,\rho ,d)/(C,\alpha ,\rho ,d)$ type-I assumptions. By citing a nontrivial example, it is shown that a second order $(C,\alpha ,\rho ,d)$ type-I function need not be $(F,\alpha ,\rho ,d)$ type-I. Several known results are obtained as special cases. References Ahmad, I., Husain, Z., Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity. J. Optimiz. Theory Appl. 129:255–275, 2006. doi:10.1007/s10957-006-9057-0 Ahmad, I., Husain, Z., Sharma, S., Second-order duality in nondifferentiable minmax programming involving type-I functions. J. Comput. Appl. Math. 215:91–102, 2008. doi:10.1016/j.cam.2007.03.022 Antczak, T., Generalized fractional minimax programming with $B$-$(p, r)$-invexity. Comput. Math. Appl. 56:1505–1525, 2008. doi:10.1016/j.camwa.2008.02.039 Chinchuluun, A., Yuan, D. H., Pardalos, P. M., Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154:133–147, 2007. doi:10.1007/s10479-007-0180-6 Du, D.-Z., Pardalos, P. M., Minimax and applications, Kluwer Academic Publishers, Dordrecht, 1995. http://vlsicad.eecs.umich.edu/BK/Slots/cache/www.wkap.nl/prod/b/0-7923-3615-1 Hachimi, M., Aghezzaf, B., Second order duality in multiobjective programming involving generalized type I functions. Numer. Funct. Anal. Optimiz. 25:725–736, 2005. doi:10.1081/NFA-200045804 Husain, Z., Ahmad, I., Sharma, S., Second order duality for minmax fractional programming. Optimiz. Lett. 3:277–286, 2009. doi:10.1007/s11590-008-0107-4 Hu, Q., Yang, G., Jian, J., On second order duality for minimax fractional programming. Nonlinear Anal. 12:3509–3514, 2011. doi:10.1016/j.nonrwa.2011.06.011 Lai, H. C., Lee, J. C., On duality theorems for a nondifferentiable minimax fractional programming. J. Comput. Appl. Math. 146:115–126, 2002. doi:10.1016/S0377-0427(02)00422-3 Lai, H. C., Liu, J. C., Tanaka, K., Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 230:311–328, 1999. doi:10.1006/jmaa.1998.6204 Liu, J. C., Wu, C. S., On minimax fractional optimality conditions with invexity. J. Math. Anal. Appl. 219:21–35, 1998. doi:10.1006/jmaa.1997.5786 Long, X. J., Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with $(C,\alpha ,\rho ,d)$-convexity. J. Optimiz. Theory Appl. 148:197–208, 2011. doi:10.1007/s10957-010-9740-z Schmitendorf, W. E., Necessary conditions and sufficient conditions for static minmax problems. J. Math. Anal. Appl. 57:683–693, 1977. doi:10.1016/0022-247X(77)90255-4 Sharma, S., Gulati, T. R., Second order duality in minmax fractional programming with generalized univexity. J. Glob. Optimiz. 52:161–169, 2012. doi:10.1007/s10898-011-9694-1 Yuan, D. H., Liu, X. L., Chinchuluun, A., Pardalos, P. M., Nondifferentiable minimax fractional programming problems with $(C,\alpha , \rho , d)$-convexity. J. Optimiz. Theory Appl. 129:185–199, 2006. doi:10.1007/s10957-006-9052-

Minmax fractional programming problem involving generalized convex functions

In the present study we focus our attention on a minmax fractional programming problem and its second order dual problem. Duality results are obtained for the considered dual problem under the assumptions of second order $\left( {F,\alpha ,\rho ,d}\right)$ -type I functions

Optimizing Measures of Risk: A Simplex-like Algorithm

The minimization of general risk or dispersion measures is becoming more and more important in Portfolio Choice Theory. There are two major reasons. Firstly, the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several risk or dispersion measures. The representation theorems of risk measures are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual (and therefore the primal) problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed