Duality in vector optimization. I: Abstract duality scheme. (English) Zbl 0556.49010

Tran Quoc Chien

Kybernetika 20, 304-313 (1984).

The paper was conceived by the author in three parts. The first part presents an abstract duality scheme. The second one discusses dual problems in vectorial quasiconvex programming. The last part is devoted to fractional programming. In the following a summary of the first part is presented: Let Y be a topological linear space. For a set \(A\subset Y\) the notions \(Sup^ s\), \(Inf^ w\), \(Max^ s\) and \(Min^ w\) are defined. Let E be a nonempty set and \((\Lambda_*,\Lambda^*)\) an ordered interval from Y and let \(\{P,Q_ y\); \(y\in (\Lambda_*,\Lambda^*)\}\) be a system of subsets belonging to E so that: For all \(y',y''\in (\Lambda_*,\Lambda^*):\) \(y'\leq y''\Rightarrow Q_{y''}\subset Q_{y'}\). We define \(Q=\cup_{y\in (\Lambda_*,\Lambda^*)}Q_ y\), \(P_ 0=P\cap Q\). For a set \(E^*\in \exp E\) we define the sets \(P^*=\{F\in E^*|\) \(P\subset F\}\) and \(Q^*_ y=\{F\in E^*|\) \(Q_{y'}\cap F=\emptyset\), \(\forall y'\geq y\}\). We also define \(Q^*=\cap_{y\in (\Lambda_*,\Lambda^*)}Q_ y\), \(P^*_ 0=P^*\cap Q^*\), \(\mu (a)=\{y\in (\Lambda_*,\Lambda^*)|\) \(a\in Q_ y\}\), \(\nu (F)=\{y\in (\Lambda_*,\Lambda^*)|\) \(F\in Q^*_ y\}\) and consider the following pairs of optimization problems: \[ \text{''find}\quad Sup^{s(w)}\cup_{a\in P_ 0}\mu (a)=S_ p^{s(w)}\text{''\quad and\quad ''find}\quad Inf^{s(w)}\cup_{F\in P_ 0^*}\nu (F)=I_ d^{s(w)}\text{'',} \]
\[ \text{ ''find}\quad Max^{s(w)}\cup_{a\in P_ 0}\mu (a)=M_ p^{s(w)}\text{''\quad and\quad ''find}\quad Min^{s(w)}\cup_{F\in P^*_ 0}\nu (F)=M_ d^{s(w)}\text{''.} \] The author establishes duality principles for the two pairs of dual problems.

Reviewer: S.Mititelu

MSC:

49N15	Duality theory (optimization)
46A40	Ordered topological linear spaces, vector lattices
90C48	Programming in abstract spaces
90C30	Nonlinear programming
49J27	Existence theories for problems in abstract spaces

Keywords:

abstract duality scheme; duality principles

References:

[1]	G. S. Rubinstein: Duality in mathematical programming and some question of convex analysis. Uspehi Mat. Nauk 25 {155) (1970), 5, 171- 201. In Russian.}
[2]	V. V. Podinovskij, V. D. Nogin: Pareto Optimal Solutions in Multiobjective Problems. Nauka, Moscow 1982. In Russian. · Zbl 0496.90053
[3]	T. Tanino: Saddle points and duality in multi-objective programming. Internat. J. Systems Sci. 13 (1982), 3, 323-335. · Zbl 0487.49021 · doi:10.1080/00207728208926352
[4]	J. W. Niewehuis: Supremal points and generalized duality. Math. Operationsforsch. Statist. Ser. Optim. 11 (1980), 1, 41-59. · Zbl 0514.90073
[5]	T. Tanino, Y. Sawaragi: Duality theory in multiobjective programming. J. Optim. Theory Appl. 27 (1979), 4, 509-529. · Zbl 0378.90100 · doi:10.1007/BF00933437
[6]	T. Tanino, Y. Sawaragi: Conjugate maps and duality in multiobjective programming. J. Optim. Theory Appl. 31 (1980), 4, 473-499. · Zbl 0418.90080 · doi:10.1007/BF00934473
[7]	S. Brumelle: Duality for multiobjective programming convex programning. Math. Oper. Res. 6 (1981), 2, 159-172. · Zbl 0497.90068 · doi:10.1287/moor.6.2.159
[8]	D. Wolfe: A duality theorem for nonlinear programming. Quart. Appl. Math. 19 (1981), 239-244. · Zbl 0109.38406
[9]	M. Schechter: A subgradient duality theorem. J. Math. Anal. Appl. 61 (1977), 850-855. · Zbl 0369.90104 · doi:10.1016/0022-247X(77)90183-4
[10]	R. T. Rockafellar: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12 (1974), 2, 268-285. · Zbl 0257.90046 · doi:10.1137/0312021
[11]	S. Schaible: Fractional programming I, duality. Management Sci. 23 (1976), 8, 858-867. · Zbl 0338.90050 · doi:10.1287/mnsc.22.8.858
[12]	S. Schaible: Duality in fractional programming: a unified approach. Oper. Res. 24 (1976), 452-461. · Zbl 0348.90120 · doi:10.1287/opre.24.3.452