Introduction

We treat the primal mixed semidefinite-quadratic-linear program (SQLP):

where is a block diagonal symmetric matrix variable, with block sizes , , , respectively, each greater or equal to two; is a block vector variable, with block sizes respectively, each greater or equal to two; and is a vector of length . The quantities and , , are block diagonal matrices. The quantities and , are block vectors, and and , , are vectors. All vectors are column vectors. The quantity is the trace inner product ( ), i.e. .

Each of the three inequalities in this primal program has a different meaning, each corresponding to a different kind of cone:

• The first kind of inequality is the semidefinite constraint. means that the matrix is positive semidefinite, or equivalently that each of its blocks is positive semidefinite.
• The second kind of inequality describes the quadratic cone constraints. Writing for brevity, with block structure

   

  where

  

  the constraint , i.e. means that, for each block i,

   

  Any convex quadratic constraint can be converted to this form.

• The third kind of inequality is the standard one: means each component of is nonnegative.

Thus the feasible set is a product of semidefinite, quadratic and nonnegative orthant cones, intersected with m hyperplanes. It is possible that one or more of the three parts of the SQLP is not present, i.e., any of s (the number of blocks in ), q (the number of quadratic blocks in ), or (the length of ) may be zero. If q=0, the SQLP reduces to an ordinary SDP and if s=0 the SQLP reduces to QCLP (convex quadratically constrained linear programming).

The standard form given here is a very convenient one. The dual SQLP is

Note again the three different kinds of inequalities on the three dual slack variables. In control applications, the first of the three is usually called a linear matrix inequality (LMI), since it can also be written .

We shall use the following notation. Let

   

where denotes the Frobenius matrix norm. Assuming the existence of a strictly feasible primal or dual point, it is well known that the optimality conditions may be expressed by the primal feasibility equation , the dual feasibility equation , and the complementarity condition (together with the inequality constraints). We will also wish to refer to the quantities