# The Scope of Linear Matrix Inequality Techniques

Thursday, November 5, 2015 - 11:00am - 12:00pm

Lind 305

J. William Helton (University of California, San Diego)

One of the main developments in optimization over the last 20 years is

Semi-Definite Programming. It treats problems which can be expressed as a

Linear Matrix Inequality (LMI). Any such problem is necessarily convex,

so the determining the scope and range of applicability comes down to the

question:

How much more restricted are LMIs than Convex Matrix Inequalities?

The talk gives a survey of what is known on this issue and will be

accessible to about anybody.

There are several main branches of this pursuit.

First there are two fundamentally different classes of linear systems

problems. Ones whose statements do depend on the dimension of the

system explicitly and ones whose statements do not.

Dimension dependent systems problems lead to traditional

semialgebraic geometry problems, while dimension free systems

problems lead directly to problems in matrix unknowns and a new area

which might be called noncommutative semialgebraic geometry.

Most classic problems of control lead to noncommutative problems.

In this talk after laying out the distinctions above

we give results and conjectures on the answer to

the LMI vs convexity question.

Semi-Definite Programming. It treats problems which can be expressed as a

Linear Matrix Inequality (LMI). Any such problem is necessarily convex,

so the determining the scope and range of applicability comes down to the

question:

How much more restricted are LMIs than Convex Matrix Inequalities?

The talk gives a survey of what is known on this issue and will be

accessible to about anybody.

There are several main branches of this pursuit.

First there are two fundamentally different classes of linear systems

problems. Ones whose statements do depend on the dimension of the

system explicitly and ones whose statements do not.

Dimension dependent systems problems lead to traditional

semialgebraic geometry problems, while dimension free systems

problems lead directly to problems in matrix unknowns and a new area

which might be called noncommutative semialgebraic geometry.

Most classic problems of control lead to noncommutative problems.

In this talk after laying out the distinctions above

we give results and conjectures on the answer to

the LMI vs convexity question.