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## Download Encyclopedia of Optimization by Claire S. Adjiman, Christodoulos A. Floudas (auth.), PDF

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2b). 3 is based on a result presented in , which uses the lower vertex matrix H, such that (H)ij - hij, and the upper vertex matrix H, such (H)ij - hij. The minimum eigenvalue of [H] is such that Amin ([H]) > Amin(H) - where the coefficients of A depend on the elements of the interval Hessian matrix [HI. A lower bound on the roots of this polynomial can then obtained by calculating the minimum roots of only four real polynomials. The appropriate bounding polynomials are the Kharitonov polynomials if i-j, ifuiuj>_0, i~j, ( hij ifuiuj <0, i~j, where all possible combinations of the signs of the arbitrary scalars ui and uj are enumerated.

If si ~ O, then go to C). If si - 0 and T -- v ~ r i -- O, then set xi+l - xi, H i + l = H i and go to F), else stop (the system has no solution). c) Compute the search vector Pi by (4) Pi - H [ z i , where zi E R n is arbitrary save for the condition v[AH[zi (5) ¢ O. D) Update the estimate of the solution by (6) Xi+l - xi - aiPi, where the stepsize (~i is given by vTi ri (7) c~i = r [ A p i " E) Update the matrix Hi by Hi+l - Hi- giATviw[gi w[giATvi (8) ' where wi E R n is arbitrary save for the condition w(HiATvi ¢ 0.

The form given by (11) can be used to construct convex underestimators for the objective function and inequality constraints. Equality Constraints. For nonlinear equality constraints, two different convexification/relaxation schemes are used, depending on the mathematical structure of the function. If the equality h(x - 0 involves only linear, bilinear, trilinear, fractional and fractional trilinear terms, it is first decomposed into the equivalent equality constraint bt (12) cTx q- E bixBi,lXB,,2 i=l tt + E + X F~,2 +Z = o, X FT~,3 i=1 bt tt c T x + E biwB, -t- E tiwT, i=-I i=l It ftt + i=1 B r a n c h i n g V a r i a b l e Selection.

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