Global methods for optimizing over the efficient set
Let X{sub E} denote the efficient set of the multiple objective linear programming problem VMAX Cx subject to x {element_of} X, where C is a p {times} n matrix and X is a polytope in R{sup n}. We consider the problem max {l_brace}dx : x {element_of} X{sub E}{r_brace} which has many applications in multiple decision making. Since X{sub E} is generally nonconvex, this problem is classified as a global optimization problem. It has been shown that there exists a simplex {Lambda} in R{sup p} such that X{sub E} = {l_brace} {element_of} X : g{lambda} - {lambda}Cx {<=} 0, {lambda} {element_of} {Lambda}{r_brace} where g{lambda} = sup {l_brace}Cy : y {element_of} X{r_brace}. We seek a global {epsilon}-optimal solution {bar x} such that dx {<=} d{bar x} + {epsilon}, {forall}x {element_of} X{sub E}. In our approach, the main problem is to check, for a given {alpha}, whether the set X{sub E}{sup {alpha}} = {l_brace}x {element_of} X{sub E} : dx {>=} {alpha}{r_brace} = {l_brace}x {element_of} : g{lambda} - {lambda}Cx {<=} 0, {lambda} {element_of} {Lambda}, dx {>=} {alpha}{r_brace} is empty or not. We show that this problem can be reduced to the problem of finding a point of the set D/C where D, C are convex sets in R{sup p} {times} R. Based on this fact we propose different algorithms for obtaining a global {epsilon}-optimal solution, namely (1) an Outer Approximation algorithm; (2) a Bisection Search algorithm; (3) a Branch and Bound algorithm. All these algorithms are implementable and require solving only linear programs. Moreover, they are efficient when the number of criteria p is small relative to n.
- OSTI ID:
- 36638
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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