一、 課程背景與目的:

The second-order cone programs (SOCP) have been an attraction due to plenty

of applications in engineering and finance. To deal with this special type

of optimization problems involved second-order cone (SOC). We believe that

a few items: (i) spectral decomposition associated SOC, (ii) analysis of

SOC functions, (iii) SOC-convexity and SOC-monotonicity are crucial concepts.

In this short course, we go through all these concepts and try to provide the

readers a whole picture regarding SOC functions and their applications.

As will be introduced, the SOC functions are indeed vector-valued

functions associated with SOC, which are accompanied by Jordan product.

However, unlike the matrix multiplication, the Jordan product associated

with SOC is not associative which is the main source of difficulty when

we do the analysis. Therefore, the ideas for proofs are usually quite

different from those for matrix-valued functions. In other words, although

SOC and positive semidefinite cone both belong to symmetric cones, the

analysis for them are different. In general, the arguments are more tedious

and need subtle arrangements in the SOC setting. This is due to the feature

of SOC.

To deal with second-order cone programs (SOCPs) and second-order cone

complementarity problems (SOCCPs), many methods rely on some SOC

complementarity functions or merit functions to reformulate the KKT

optimality conditions as a nonsmooth (or smoothing) system of equations

or an unconstrained minimization problem. In fact, such SOC complementarity

or merit functions are comprised of SOC functions. In other words,

the vector-valued functions associated with SOC are heavily used in the

solutions methods for SOCP and SOCCP. Therefore, further study on

these functions will be helpful for developing and analyzing more

solutions methods.

 

二、課程之大綱與講者:

由臺灣師大數學系陳界山教授主講,課程規劃為五次。

Lecturer: Prof. Jein-Shan Chen, Department of Mathematics,

 

National Taiwan Normal University

 

The structure of the short course is as follows.(課程大綱如下)

 

1. SOC functions

2. SOC-convex functions and SOC-monotone functions

3. Algorithmic applications

4. SOC means applications

5. Possible extensions