This paper concerns nonlinear semidefinite programming problems for which n
o convexity assumptions can be made. We derive first- and second-order opti
mality conditions analogous to those for nonlinear programming. Using techn
iques similar to those used in nonlinear programming, we extend existing th
eory to cover situations where the constraint matrix is structurally sparse
. The discussion covers the case when strict complementarity does not hold.
The regularity conditions used are consistent with those of nonlinear prog
ramming in the sense that the conventional optimality conditions for nonlin
ear programming are obtained when the constraint matrix is diagonal.