Constrained convex optimization problems – well-posedness and stability. (English) Zbl 0830.90119

A constrained convex optimization problem is under consideration. The constraints are given in the form of an inequality in a partially ordered Banach space. The convexity of the constraint function is defined with respect to the partial order. Four types of stability for such kind of problems are investigated: (a) Hadamard well-posedness; (b) strong well- posedness; (c) Levitin-Polyak well-posedness; (d) Tikhonov well- posedness.
The principal result says that if both the objective functional and the constraint function are convex, then the following chain of implications is true: \((a)\to (b)\to (c)\to (d)\). Moreover, all the types of well- posedness are mutually equivalent under the Slater condition.
Withdrawal of the assumption of uniqueness of the minimum point leads to generalized concepts of stability. It is obtained that for generalized (a), (b), (c) and (d) well posedness the above-mentioned result holds true as well.
The minimality of the assumptions is confirmed by counter-examples.
Reviewer: D.Silin (Graz)


90C25 Convex programming
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI


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