Linear Matrix Inequalities
DefinitionPositive Semi-Definite MatricesRecall from here that a An alternative condition for ![]() The set of PSD matrices is convex, since the conditions above represent (an infinite number of) ordinary linear inequalities on the elements of the matrix Examples:
Standard formA linear matrix inequality is a constraint of the form ![]() where the
The matrices An alternate form for LMIs is as the intersection of the positive semi-definite cone with an affine set: ![]() where LMIs and Convex SetsLet us denote by ![]() The set ![]() Since ![]() with
Multiple LMIsWe can combine multiple LMIs into one. Consider two affine maps from ![]() are equivalent to one LMI, involving a larger matrix of size ![]() This corresponds to intersecting the two LMI sets. Special CasesLMIs include as special cases the following. Ordinary affine inequalitiesConsider single affine inequality in ![]() where Using the result above on multiple LMIs, we obtain that the set of ordinary affine inequalities ![]() can be cast as a single LMI ![]() Second-order cone inequalitiesSecond-order cone (SOC) inequalities can be represented as LMIs. To see this, let us start with the ‘‘basic’’ SOC ![]() Indeed, we check that for every ![]() if and only if More generally, a second-order cone inequality of the form ![]() with ![]() The proof relies on the Schur complement lemma. |