M. Minoux. Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....direction for q. When the size of # is polynomial in n (eg d = 1) we are able, for any u u u , to compute q(u u u) and a subgradient at u u u in polynomial time. Since q is concave, DCAP and therefore CAP can be solved in polynomial time by the ellipsoid algorithm of Shor Khachian [16, 23, 21, 12]. Remark 3. By perturbating the cost function f i and replacing them by f i i# for some small # 0, we can build an instance of the problem such that there is no cell of the common support #(f 0 , f i i#, f n n#) on which f i f j (i j)# is constant. As a consequence, D ....

....cost of all the sequenced task in s. That proves that by adding the constant #(P ) to the minimum of 1 CAP, we get a lower bound for s, that is for any element of S. 4. 3 Implementation and computational results We implemented several subgradient algorithms among them presented in Minoux book [16]. The one that seems the most e#cient is Shor s method with space dilatation [20] We used several classical tricks in order to limit the computation time of the lower bound : At each iteration of the subgradient method, the value q(u u u) is a lower bound of the optimum of 1 CAP so it is a ....

M. Minoux. Mathematical Programming: Theory and Algorithms. Wiley & Sons, 1986.

....for each mobile does not depend on the power allocation for other mobiles in problem (B) while the utility function for each mobile depends on the power allocation for all mobiles in problem (A) 0 7803 7476 2 02 17.00 (c) 2002 IEEE. To solve problem (B) we state the following result from [13] (page 231) that gives us optimality condition for general optimization problem. Lemma 1: Let f : R R and g i : R R, i = K be arbitrary functions. We define L(x, #) f(x) # g(x) w( #) m1 x L(x, #) x L(x, #) w( where x = x 1 ,x 2 , x N ) # = # 1 ,# 2 , ....

....there exists no power allocation such P # and, thus, P # is a Pareto optimal power allocation. The power allocation algorithm can be implemented in several ways. If we consider problem (F) we can use a simple line search algorithm such as a bisect algorithm and a golden section algorithm [13]. If we consider problem (G) we can use a gradient based algorithm [4] or a penalty based algorithm [3] since problem (G) is equivalent to the following convex programming problem. H) m1 P i (# i ) 1, where P i (# i . Thus, U i (# i (P i ) is a concave function for P i ....

M. Minoux, Mathematical programming:theory and algorithms, Wiley, 1986.

....now focus on the minimization of E(A; W ) with respect to W only subject to the n constraints h i (A; W ) 0, with respect to W only. This problem is well posed because it is a minimization of a convex function subject to convex constraints. Therefore using the classical Kuhn and Tucker s theorem [10], if a solution exists, the minimization of E(A; W ) with respect to W is equivalent to the search of the unique saddle point of the Lagrange function of the problem: LR (A; W; i ) 1 2 i=n X i=1 (w i ) i (w i ( X t i A y i ) 2 ) where i are Kuhn and Tucker multipliers ( ....

M. Minoux. Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986.

.... The Lagrangian associated with the definition of G S reads L (f i g i2S c ; q; fq i g i2S c ) X S c H i ( i ) Gamma q X S c i Gamma c X S OE i X S c q i ( i Gamma OE i ) At the optimum, H 0 i ( i ) q Gamma q i , and by the Kuhn Tucker theorem (see e.g. Minoux [7]) q i 0; i OE i ) q i = 0 There is therefore a subset of indices I ae S c such that G S ( X I H i (OE i ) X S c nI H i ( i ) with 8 : P I OE i P S c nI i = c Gamma P S OE i i 2 I ) H 0 i (OE i ) q i 2 S c n I ) H 0 i ( i ) q If the set of ....

M. Minoux (1986), Mathematical Programming: Theory and Algorithms, Wiley, Chichester.

....to consider a canonical hyperplane [2] where w is constrained by min r i 2R jw T r i j = 1 (13) De ne the integer set I R 4 = fi : r i 2 R g and the class indicator y i = 1; 8r i 2 R . The maximization of the margin (12) with the constraint (13) using the classical Lagrangian theory [4] gives rise to the optimal separating hyperplane: wSVM = X i2I R g i y i r i (14) where g = arg min g 1 2 X i2I R X j2IR g i g j y i y j r T i r j X i2I R g i (15) g i 0; 8i 2 I R (16) The optimization problem (15) with (16) is a convex quadratic programming, whose ....

....j2IR g i g j y i y j r T i r j X i2I R g i (15) g i 0; 8i 2 I R (16) The optimization problem (15) with (16) is a convex quadratic programming, whose solution g can be computed eciently. As g i are the Lagrange multipliers of the primal 3 problem, from the Kuhn Tucker conditions [4] g i y i w T SVM r i 1 = 0 (17) only those r i , which satisfy y i w T SVM r i = 1, will have non zero Lagrange multipliers. These points are the support vectors (SVs) 2] All the SVs lie on the margin and the number of SVs can be very small. The hyperplane wSVM is uniquely ....

M. Minoux, Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986. S. Chen, A.K. Samingan and L. Hanzo (Department of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K.) Tel./Fax: +44 (0)23 8095 6660/4508; E-mail: sqc@ecs.soton.ac.uk.

....# S . The Lagrangian associated with the definition of GS reads L ( # j j#S c , q, q j j#S c ) X S c H j (# j ) q X S c # j c X S # j X S c q j (# j # j ) At the optimum, H # j (# j ) q q j , and by the Kuhn Tucker theorem (see e.g. Minoux [9]) q j # 0, # j # j # q j = 0. There is therefore a subset of indices I # S c such that GS ( X I H j (# j ) X S c I H j (# j ) with 8 : P I # j P S c I # j = c P S # j , j # I # H # j (# j ) # q, j # S c I # H # j (# j ) q. We then ....

M. Minoux (1986), Mathematical Programming: Theory and Algorithms, Wiley, Chichester.

....of feasible routes, a multicommodity flow problem with flow between o d pair (i; j) equaling P k 2 ij C k (i; j) must be solved. 10 For a review of the multicommodity flow problem and its numerous variations and computationally efficient solution techniques refer to the textbook by Minoux [43]. Now consider the case where the underlying topology G[X;U ] is given and the capacities F u ; u 2 U , have to be chosen so as to design a minimal cost physical network. The multicommodity flow problem can once again be used to design a network once again if the cost of each link is a linear ....

M. Minoux. Mathematical Programming Theory and Algorithms. John Wiley and Sons, 1986.

....in general, the surrogate models, the nonlinear barrier patrol model, and the constraints yield a nonlinear mixed integer programming problem. Although there are various approaches to solving difficult mathematical programming problems of this and similar types [e.g. Bazaraa and Shetty, 1979; Minoux, 1986], the emphasis is typically on optimality. As noted above, this emphasis of operations research is not the only aim of candle lighting. While the mathematically optimal solution is always of interest, we are also always interested in insight, much of which comes from exploring other solutions. ....

Minoux, Michel., 1986. Mathematical programming : theory and algorithms, Wiley, New York.

.... a canonical hyperplane [11] where w is constrained by min r i 2R jw T r i j = 1 : 27) De ne the integer set I R 4 = fi j r i 2 Rg ; 28) and the class indicator y i = 1; 8r i 2 R : 29) The maximisation of the margin (26) with the constraint (27) using the classical Lagrangian theory [22], 23] gives rise to the optimal separating hyperplane: wSVM = X i2I R g i y i r i ; 30) where g = arg min g 1 2 X i2I R X j2IR g i g j y i y j r T i r j X i2I R g i ; 31) 8 g i 0; 8i 2 I R : 32) The optimization problem (31) with (32) is a standard quadratic programming, ....

.... wSVM = X i2I R g i y i r i ; 30) where g = arg min g 1 2 X i2I R X j2IR g i g j y i y j r T i r j X i2I R g i ; 31) 8 g i 0; 8i 2 I R : 32) The optimization problem (31) with (32) is a standard quadratic programming, whose solution g can readily be computed eciently [22], 23] Notice that g i are the Lagrange multipliers of the primal problem. From the Kuhn Tucker conditions [22] 23] g i y i w T SVM r i 1 = 0 ; 33) and only those points r i , which satisfy y i w T SVM r i = 1, will have non zero Lagrange multipliers. These points are termed ....

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M. Minoux, Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986.

.... at a maximum of L over the orthant x; y; z 0, the following conditions hold: s = U 0 s (x s ) if x s 0 U 0 s (x s ) if x s = 0 (4) s(r) X j2r j if y r 0 X j2r j if y r = 0 (5) j = 0 if z j 0 0 if z j = 0 (6) From the general theory of constrained convex optimization ([8], chapter 5; 13] chapter 3) it follows that there exists a quadruple ( x; y) which satisfies U 0 (x) Hy = x; Gamma Gamma U 0 (x) Delta T x = 0 (7) 0; Ax C; T (C Gamma Ax) 0 (8) T H T A; y 0; T A Gamma T H)y = 0 (9) and that, further, the vector x then ....

M. Minoux, Mathematical Programming: Theory and Algorithms. Wiley, Chichester, 1986.

....# #H #1 ## #s# # ##J## #s# # ### #31# where # is the learning rate and # # 1. Note that the learning rate is unnecessary in the standard Newton Raphson method. Here, the learning rate is adopted to speed up learning so that # #s#1# can be still in the neighborhood of # #s# for convergence (Minoux, 1986). Like the IRLS algorithm (Jordan Jacobs, 1994) it is easy to generalize the derivation to allow fixed weights to be associated with data pairs. As mentioned before, such a generalization is necessary for the ME architecture with the EM algorithm since the EM algorithm defines observation ....

....results also indicate that the BFGS algorithm does not yield significantly faster learning for problems studied in this paper. In particular, the BFGS algorithm is often sensitive to the learning rate though the high precision linear search is not necessary theoretically (Fletcher, 1987; Minoux, 1986). Our simulation results are K. Chen et al. Neural Networks 12 (1999) 1229 1252 1250 . Combination of weighted results X x 1 xt x T . Network Gating S Gating Network Expert Network 1 Expert Network N O(X) X . Fig. 8. The architecture of a modified mixture of experts for speaker ....

[Article contains additional citation context not shown here]

Minoux, M. (1986). Mathematical programming: theory and algorithms, New York: Wiley.

.... problem for the Quad placement machine, and sub optimal solutions were obtained by dividing the problem into two sub problems where the placement sequence and component feeder assignments were optimized separately by using TSP and minimum weight maximum matching algorithms, respectively [8, 7, 10]. Both the simulation results and the machine test results showed the effectiveness of their algorithm. Although this model can be extended to other machines having similar mechanism, it can not be used in machines like FUJI, where both the board and the device moves simultaneously and components ....

....case well, where more 14 than 50 of the 867 components belong to the 6 of the 95 types, and the time reduction is most prominent. 7 Conclusion Many optimization problems in printed circuit board assembly and other manufacturing applications can be formulated as mathematical programming problem [10], and can be transformed to or approximated by some standard operations research problems such as TSP [8] or minimum weight maximum match [7] etc. and existing algorithms and software can be used to solve these problems. The FUJI FCP IV machine placement optimization problem presented in this ....

M. Minoux. Mathematical Programming: Theory and Algorithms. John Wiley and Sons, 1986.

....is independent of . In this case, the formula in (3.7) reduces to the result in [3, 9] However for 6= 1, Z is in general a function of 6= 0, and thus computation of fl opt is not that simple. In the remainder of the section, we outline a simple algorithm that employs the Newton Raphson method [8] for computing fl opt . The following result is the key. 6 Lemma 3.3 Let Y 2 and Z( be the stabilizing solutions to AREs (2.7) and (3.6) respectively. Then the function defined by f( ae(Y 2 Z( ae(Y 2 Z ( is monotonically nondecreasing with respect to , and a concave function of ....

....and Z from (3.6) ffl Step 2: Let ffl 0 be the error tolerance. For i = 0; 1; 2; do the following: a) Set = i and compute the stabilizing solution Z( from (3.6) and Z 0 ( from (3.8) b) Compute f( and f 0 ( according to (3. 11) c) Compute i 1 using the Newton Raphson method [8]. d) If j Gamma i 1 j ffl, set fl opt = 1 Gamma f( Gamma1=2 . Stop; Otherwise set i : i 1. Go to Step 2. The following result shows the convergence property of the proposed iterative algorithm. 7 Theorem 3.4 Suppose that the realization of P (s) is both stabilizable and ....

M. Minoux, Mathematical Programming -- Theory and Algorithms, John Wiley and Son's, 1986.

.... consider a canonical hyperplane [3] where w is constrained by min r i 2R jw T r i j = 1 : 13) Define the integer set I R 4 = fi : r i 2 Rg and the class indicator y i = 1; 8r i 2 R ( 1) The maximization of the margin (12) with the constraint (13) using the classical Lagrangian theory [5] gives rise to the optimal separating hyperplane: wSVM = X i2IR g i y i r i ; 14) where g = arg min g 1 2 X i2IR X j2IR g i g j y i y j r T i r j X i2IR g i ; 15) g i 0; 8i 2 I R : 16) The optimization problem (15) with (16) is a quadratic programming problem, whose ....

....X j2IR g i g j y i y j r T i r j X i2IR g i ; 15) g i 0; 8i 2 I R : 16) The optimization problem (15) with (16) is a quadratic programming problem, whose solution g can be computed efficiently. As g i are the Lagrange multipliers of the primal problem, from the Kuhn Tucker conditions [5] g i y i w T SVM r i 1 = 0 ; 17) only those points r i , which satisfy y i w T SVM r i = 1, will have non zero Lagrange multipliers. These points are the SVs [3] All the SVs lie on the margin and the number of SVs can be very small. Let RSV be the set of SVs. The hyperplane wSVM is ....

M. Minoux, Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986.

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M. Minoux. Mathematical Programming: Theory and Algorithms. Chichester: John Wiley and Sons, 1986.

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Minoux, M. (1986). Mathematical programming: Theory and algorithms. New York: Wiley.

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Minoux, M. (1986). Mathematical programming: theory and algorithms, New York: Wiley.

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Minoux, M. (1986). Mathematical Programming: Theory and Algorithms, John Wiley & Sons.

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M. Minoux, Mathematical Programming: Theory and Algorithms, John Wiley & Sons, 1986.

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M. Minoux, Mathematical programming:theory and algorithms. Wiley, 1986.

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M. Minoux. Mathematical Programming: Theory and Algorithms. John Wiley and Sons, 1986.

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M. Minoux, Mathematical programming:theory and algorithms. Wiley, 1986.

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Minoux, M. (1986). Mathematical Programming: Theory and Algorithms, John Wiley & Sons.

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M. Minoux, Mathematical Programming: Theory and Algorithms (J. Wiley, New York, 1986).

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