Strong duality in conic linear programming: facial reduction and extended duals

The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program $$(P) \sup {<c, x> | Ax \leq_K b}$$ in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana's dual using certificates from a facial reduction algorithm. Here we give a clear and self-contained exposition of facial reduction, of extended duals, and generalize Ramana's dual: -- we state a simple facial reduction algorithm and prove its correctness; and -- building on this algorithm we construct a family of extended duals when $K$ is a {\em nice} cone. This class of cones includes the semidefinite cone and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of a facial reduction algorithm and extended dual systems", technical report, Columbia University, 2000, available from http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of Jonathan Borwein's 60th birthday, 201

Analytical, experimental and numerical study of a graded honeycomb structure under in-plane impact load with low velocity

Given the significance of energy absorption in various industries, light shock absorbers such as honeycomb structure under in-plane and out-of-plane loads have been in the core of attention. The purpose of this research is the analyses of graded honeycomb structure (GHS) behaviour under in-plane impact loading and its optimisation. Primarily, analytical equations for plateau stress and specific energy are represented, taking power hardening model (PHM) and elastic–perfectly plastic model (EPPM) into consideration. For the validation and comparison of acquired analytical equations, the energy absorption of a GHS made of five different aluminium grades is simulated in ABAQUS/CAE. In order to validate the numerical simulation method in ABAQUS, an experimental test has been conducted as the falling a weight with low velocity on a GHS. Numerical results retain an acceptable accordance with experimental ones with a 5.4% occurred error of reaction force. For a structure with a specific kinetic energy, the stress–strain diagram is achieved and compared with the analytical equations obtained. The maximum difference between the numerical and analytical plateau stresses for PHM is 10.58%. However, this value has been measured to be 38.78% for EPPM. In addition, the numerical value of absorbed energy is compared to that of analytical method for two material models. The maximum difference between the numerical and analytical absorbed energies for PHM model is 6.4%, while it retains the value of 48.08% for EPPM. Based on the conducted comparisons, the numerical and analytical results based on PHM are more congruent than EPPM results. Applying sequential quadratic programming method and genetic algorithm, the ratio of structure mass to the absorbed energy is minimised. According to the optimisation results, the structure capacity of absorbing energy increases by 18% compared to the primary model

Nonlinear local error bounds via a change of metric

International audienceIn this work, we improve the approach of the second author and V. Motreanu [Math. Program. 114 (2008), 291–319] to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and it allows dealing with local error bounds in an appropriate way. We present some consequences of the general results in the framework of classical nonsmooth analysis, involving Banach spaces and subdifferential operators. In particular, we describe connections between local quadratic growth of a function and metric regularity of its subdifferential

Optimality conditions (In Pontryagin form)

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