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

Konsulova, A. S.; Revalski, J. P.

Numer. Funct. Anal. Optimization 15, No. 7-8, 889-907 (1994).

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)

MSC:

90C25	Convex programming
49J45	Methods involving semicontinuity and convergence; relaxation

Keywords:

constrained convex optimization; inequality in a partially ordered Banach space; stability

References:

[1]	DOI: 10.1007/BF01764131 · Zbl 0769.54009 · doi:10.1007/BF01764131
[2]	DOI: 10.2307/2001800 · Zbl 0753.49007 · doi:10.2307/2001800
[3]	DOI: 10.1080/02331939008843576 · Zbl 0719.49013 · doi:10.1080/02331939008843576
[4]	DOI: 10.1007/BF01371085 · Zbl 0762.90073 · doi:10.1007/BF01371085
[5]	Berdishev V. I., USSR Math. Sbornik 103 pp 467– (1977)
[6]	DOI: 10.1007/BF01198356 · Zbl 0662.46015 · doi:10.1007/BF01198356
[7]	DOI: 10.2307/2001823 · Zbl 0681.46013 · doi:10.2307/2001823
[8]	Beer G., Solvability for constrained problems (1991)
[9]	DOI: 10.1287/moor.17.3.715 · Zbl 0767.49011 · doi:10.1287/moor.17.3.715
[10]	DOI: 10.2307/2154406 · Zbl 0810.54011 · doi:10.2307/2154406
[11]	Dontchev A., Well-posed Optimization Problems 1543 (1993) · Zbl 0797.49001
[12]	DOI: 10.1007/BF00927717 · Zbl 0177.12904 · doi:10.1007/BF00927717
[13]	Holmes R., Geometric Functional Analysis and its Applications (1975) · Zbl 0336.46001
[14]	Kuratowski K., Topology I (1966)
[15]	Levitin E. S., Soviet Math. Dokl. 7 pp 764– (1966)
[16]	Lucchetti R., Boll Un. Math. Hal, Ser, C 6 pp 337– (1982)
[17]	DOI: 10.1016/0022-247X(82)90187-1 · Zbl 0487.49013 · doi:10.1016/0022-247X(82)90187-1
[18]	DOI: 10.1016/0001-8708(69)90009-7 · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[19]	Penot J.-P, C. R. Acad. Sc. Paris 288 pp 241– (1979)
[20]	Revalski J., Comp. rend. Acad. bulg. sci. 38 pp 1431– (1985)
[21]	Revalski J., Acta Univ. Carolinae Math, et Phys. 28 pp 117– (1987)
[22]	DOI: 10.1007/BF00952826 · Zbl 0798.49031 · doi:10.1007/BF00952826
[23]	Tykhonov A. N., USSR J. Comp. Math. Math. Phys. 6 pp 631– (1966)