ON OPTIMALITY AND DUALITY IN INTERVAL-VALUED VARIATIONAL PROBLEM WITH B-(p,r)-INVEXITY

In this paper, we consider a class of interval-valued variational optimization problem. We extend the definition of B -( p,r )- invexity which was originally defined for scalar optimization problem to the interval-valued variational problem. The necessary and sufficient optimality conditions for the problem have been established under B -( p,r )-invexity assumptions. An application, showing utility of the sufficiency theorem in real-world problem, has also been provided. In addition to this, for an interval- optimization problem Mond-Weir and Wolfe type duals are presented and related duality theorems have been proved. Non-trivial examples verifying the results have also been presented throughout the paper.

Necessary and sufficient conditions for the strong local minimality of C1 extremals on a class of non-smooth domains

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky and Mengesha (2009), where smoothness assumptions are made on the boundary.

Robust Stability of Plants With Mixed Uncertainties and Quantitative Feedback Theory

Quantitative Feedback Theory (QFT) has often been criticized for lack of a rigorous mathematical theory to support its claims. Yet it is known to be a very effective design methodology. In this paper, we re-examine QFT and state several results that confirm the validity of this highly effective framework proposed by Horowitz. Also provided are some additional insights into the QFT methodology that may not be immediately apparent. We consider three important fundamental questions: (i) whether or not a QFT design is robustly stable, (ii) does a robust stabilizer exist, and (iii) does a controller assuring robust QFT performance exist. The first two are obvious precursors for synthesizing controllers for performance robustness. We give necessary and sufficient conditions that unambiguously resolve the question of robust stability under mixed uncertainty, thereby, confirming that a properly executed QFT design is automatically robustly stable. Also given is a sufficiency condition for a robust stabilizer to exist which is derived from the well known Nevanlinna-Pick theory in classical analysis. Finally, we give a sufficiency theorem for the existence of a QFT controller and deduce that when the uncertain plant set is minimum phase with no unstructured uncertainty there always exists a controller satisfying robust performance specifications in the sense of QFT.

Sufficient Fritz John optimality conditions for nondifferentiable convex programming

The Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others). This sufficiency theorem is now extended to minimization (suitably defined) of a function taking values in a partially ordered space, and to (convex) objective and constraint functions which are not always differentiable. The results are expressed in terms of subgradients.