A. M. Geoffrion, "Proper efficiency and the theory of vector maximization, " J. Math. Anal. Appl., vol. 22, pp. 618--630, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and generate ROC curves. In fact, scalar optimization methods can theoretically arrive at the same ROC curves as a multiobjective optimization. Consider the following scalar optimization problem: A1) where is an element of the space of possible parameter vectors are fixed, and . Geoffrion [33] proved the following lemma. Lemma 1 (a) If maximizes (A1) then is also Pareto optimal in the vector objective space . b) Let be a convex set and let the be convex on . Then is Pareto optimal if and only if maximizes (A1) for some and . Because the multiobjective training problem, as we have ....

A. M. Geoffrion, "Proper efficiency and the theory of vector maximization, " J. Math. Anal. Appl., vol. 22, pp. 618--630, 1968.

....the scope of this work. For a taxonomy similar to the one presented here, the reader is referred to Benson and Sayin [7] 1.2. 1 General Value Functions One of the main solution strategies for multicriteria optimization problems nowadays is the scalarization approach, first described by Geoffrion [43]. Here, one or several parameterized single objective (i.e. classical) optimization problems are solved instead of one multicriteria problem. Parameters occur because there are several functions mapping IR n , the image space of the given problem, into IR, the image space of a classical ....

....E(M;K) cl(E p (M; K) But in a numerical solution method one will usually not be able to distinguish between a subset of IR n and it s closure. As a consequence, E p (M; K) is a good approximation to E(M;K) Moreover, there exist other characterizations of the set of proper efficient points [43], by which it is clear that the set of proper efficient points consists of those efficient points for which an unbounded trade off in the function values is not possible. Note that the main assumption needed is the convexity of M . Other scalarizations exist which do not need this convexity ....

Arthur M. Geoffrion. Proper Efficiency and the Theory of Vector Maximization. Journal of Optimization Theory and Applications, 22:618--630, 1968.

....Optimization and Decision Theory 3 Extremal Pareto solutions are always Pareto solutions. Their objective function vectors are extreme points of the convex hull of ff(F )jF 2 Fg. In continuous multicriteria optimization these solutions are also called properly efficient solutions, see [4]. Even the set of extremal Pareto solutions may be very large, although in most cases its cardinality is much smaller than that of the Pareto set. Choosing a scalarizing vector is equivalent to assigning relative preferences to the Q objective functions, something decision makers may want to ....

A.M. Geoffrion. Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22:618--630, 1968.

.... Hx ew d = g (4.1) x, y, a, d 0, b = 10 a i x i = 0, i = 1,2, n d y j 3 , j = 1,2, k m. Initially, let l be a small positive number. We use the value l : Dl 0 .10 7 ) This guarantees that the initial solution is the efficient solution of problem (2. 3) see, e.g. Geoffrion [1968]) but it might still be a weakly efficient solution of the original problem (2.1) If l is small enough, the race is started from the optimum of the quadratic term or from its neighborhood. Specify Dg = Dg 0 and Dl = Dl 0 . Step 4. Moving About the Efficient Frontier (Quadratic Pareto Race) ....

Geoffrion, A. (1968), "Proper Efficiency and the Theory of Vector Maximization", Journal of Mathematical Analysis and Applications, 22, pp. 618--630.

....2 IR n with z i = minff i (x) x 2 Sg Gamma ffl i = 1; n is called the ideal (utopia) criterion vector, where the entries of ffl 2 IR n are small positive numbers. Without loss of generality we assume z = 0. We define the set of properly nondominated solutions according to (Geoffrion, 1968). A point z 2 N is called properly nondominated, if there exists M 0 such that for each i = 1; n and each z 2 Z satisfying z i z i there exists a j 6= i with z j z j and z i Gamma z i z j Gamma z j M: Otherwise z 2 N is called improperly nondominated. The set of all ....

....Follows directly from the construction of fl and Lemma 3.5. Theorem 3.7 (Convex polyhedral case in IR 2 ) Let Z IR 2 be convex and polyhedral and let z 2 N . Then there exists an oblique norm fl so that z uniquely minimizes min z2Z fl(z) min x2S fl(f (x) 5) Proof. Due to (Geoffrion, 1968), there exists a supporting line of Z at z with the normal vector w 0. Define the two vectors w 1 = ffw 1 ; w 2 ) and w 2 = w 1 ; ffw 2 ) where ff 1. Denote the line defined by the normal w 1 through z as l 1 and the line defined by the normal w 2 through z as l 2 . Take the ....

Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization.

....solution. Consequently h is a weakly efficient solution of TVO. ii) If Assumption 3.2 holds, then the cost of TVO is clearly convex. Furthermore, if h is a properly efficient solution of TVO, then by the convexity of cost and the scalarization theorem of vector optimization, see Geoffrion [4]) there exists 2 int such that h is a solution of P( By Theorem 3.1, h is in equilibrium. By Theorem 2.1(ii) and the fact that 2 int, we conclude that h is in vector equilibrium. Under Assumption 3.2, Lemma 3.1 and (3.5) the vector cost F(v) of problem TVO is also IR r convex. As in ....

A.M. Geoffrion, "Proper efficiency and the theory of vector maximization," Journal of Mathematical Analysis and Applications 22 (1968) 618-630.

....1972] A feasible point x ffi 2 X is said to be an efficient solution of the MOPLP if there is no other x 2 X such that f i (x) f i (x ffi ) i = 1; m, with at least one strict inequality. Let XE and X PE denote the set of efficient and properly efficient solutions (in the sense of [Geoffrion, 1968]) of MOPLP. Geometrically related to the concept of efficiency is the concept of level sets and level curves. Let y 2 IR m . L i (y i ) fx 2 X : f i (x) y i g; i = 1; m; L i = y i ) fx 2 X : f i (x) y i g; i = 1; m: The following theorem summarizes properties of ....

....to the concept of efficiency is the concept of level sets and level curves. Let y 2 IR m . L i (y i ) fx 2 X : f i (x) y i g; i = 1; m; L i = y i ) fx 2 X : f i (x) y i g; i = 1; m: The following theorem summarizes properties of the sets XE and X PE , see [Geoffrion, 1968], Gal, 1986] Hamacher and Nickel, 1993] and [Ehrgott et al. 1996] Theorem 2.1 1. Let x ffi 2 X and y ffi i = f i (x ffi ) i = 1; m. x ffi 2 XE iff m i=1 L i (y ffi i ) m i=1 L i = y ffi i ) 1) 2. XE = X PE : 3. The set XE is connected. In the ....

Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22:618--630.

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Geoffrion A (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications 22: 618-630.

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Geoffrion, A.M. (1968), "Proper efficiency and the theory of vector maximization", Journal of Mathematical Analysis and Applications 22, 618--630.

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Geoffrion, A.M., Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618-630, 1968.

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