This paper considers the optimal induced L2gain estimator design problem for infinite dimensional systems with finite dimensional outputs. It is shown that this problem is equivalent to a dual regulator design problem. Moreover the relationship between the dual problem and a two player differential game is established. A solution to the latter problem is derived which, in turn solves the dual r...

Background Phase contrast MRI has limited practical value in clinical applications due to its inherently long scan times [Kilner,JCMR’07]. Parallel imaging[Pruessmann, MRM’01] and spatio-temporal constrained reconstruction techniques[Tsao,MRM’03] have been shown beneficial[Baltes,MRM’05]. Dual slice approaches have shown advantages as compared to standard SENSE when multiple and sufficiently se...

We develop a randomized approximation algorithm for the size of set union problem |A1∪A2∪...∪Am|, which is given list sets A1,...,Am with approximate mi Ai mi∈((1−βL)|Ai|, (1+βR)|Ai|), and biased random generators probability Prob(x=RandomElement(Ai))∈[1−αL|Ai|,1+αR|Ai|] each input element x∈Ai, where i=1,2,...,m αL,αR,βL,βR∈(0,1). The ratio |A1∪A2∪...∪Am| in range [(1−ϵ)(1−αL)(1−βL),(1+ϵ)(1+αR...

Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs an...

in this paper, using the best proximity theorems for an extensionof brosowski's theorem. we obtain other results on farthest points. finally, wedene the concept of e- farthest points. we shall prove interesting relationshipbetween the -best approximation and the e-farthest points in normed linearspaces (x; ||.||). if z in w is a e-farthest point from an x in x, then z is also a-best ...

We introduce a new g-frame (singleton g-frame), g-orthonormal basis and g-Riesz basis and study corresponding notions in some other generalizations of frames.Also, we investigate duality  for some kinds of g-frames. Finally, we illustrate an example which provides a  suitable translation from discrete frames to Sun's g-frames.

In the previous lecture, in the first place, we talked about duality for general non-convex optimization. And we know that dual functions are always convex. The dual of the dual problem is also convex. In applications, the dual of the dual can be used as the convex relaxation of the primal (and we will see this explicitly in the next lecture). Then, we covered the concepts of weak duality and s...

In mathematical programming, constraint qualifications are essential elements for duality theory. Recently, necessary and sufficient constraint qualifications for Lagrange duality results have been investigated. Also, surrogate duality enables one to replace the problem by a simpler one in which the constraint function is a scalar one. However, as far as we know, a necessary and sufficient cons...