Mathematics Colloquia and Seminars

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In contrast with most local minimization algorithms which use point information to build a local model of a given function, there is in our framework a local region associated with the current iteration point by using our notion of linear bounding functions (LBF's). Our model is then built upon the regional information of the current iteration point by using the fact that a given functional form is actually a regional information to a given optimization problem, which is basically ignored and not utilized. In other words, our model reflects the regional information in the whole local region, not just point information. As a result, our quadratic model thus found can always give us a better point without using line search, and enabled us to develop a globally convergent, line search free, and superlinear/quadratic rate algorithm without evaluating Hessian. Furthermore, there are natural ways to integrate it with various existing methods, and thus they are complimentary to each other. In the presentation, we will first briefly review the basic ideas in existing methods, and then present the concept of linear lower bounding function (LLBF's). An LLBF is basically a linear function which lies below a given function over a given box and matches its function value at one corner point. We will give graphical illustrations to show how such LLBF's can be constructed for a large class of functions called factorable functions, and then demonstrate how to use them to develop the superlinear/quadratic rate algorithm mentioned earlier. If time allows, we will discuss intuitively how LBF's can be used for various parameter optimization problems. The problem can be either constrained or unconstrained. The extended factorable functions invloved can be smooth or non-differentiable.