Several Studies Of Calmness For Convex Equations With Constraint

The concept of calmness plays a key role in the optimization theory and mathematical programming.It is closely related to issues like metric subregularity of generalized equations,nondegenerate multiplier rules,optimization conditions,or existence of error bounds.Since requiring a problem to be calm is less stringent than most of the customary constraint qualifications and it gives a Lipschitz bound on the distance of perturbed solutions from the unperturbed solution,the calmness has the further advantage of being an important stability property present in variational analysis,nonlinear optimization,nonsmooth calculus etc.This thesis mainly studies concepts of calmness,global calmness and strong calmness for a convex constraint system defined by a closed convex multifunction and a closed convex subset.Several primal characterizations of calmness are provided in terms of the contingent cone in Bouligand’s sense and the tangent derivative.As application,this thesis studies the error bound of second-order cone programing problems.Second-order cone programming(SOCP)problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order cones.It has advantages not only the convex programs have as well,but also others due to the cone structure it has.Semidefinite programming(SDP)includes SOCP as a special case,linear programs,convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems.These latter problems model applications from a broad range of fields from engineering,control and finance to robust optimization and combinatorial optimization.Several equivalent conditions of SOCP having error bounds are proved in this thesis by utilizing the established results of calmness and its form is simpler thanks to the good properties SOCP has.More over,the error bound modulus is estimated accurately in this thesis as well.