R. B. Kearfott. Treating non-smooth functions as smooth functions in global optimization and nonlinear systems solvers. In G. Alefeld, A. Frommer, and B. Lang, editors, Scientic Computing and Validated Numerics, Mathematical Research, volume 90, pages 160-172, Berlin, 1996. Akademie Verlag. 9  Home/Search   Document Details and Download   Summary   Related Articles   Check

....used to better model the real world. ffl In addition to usual binary arithmetic (addition, subtraction, multiplication, and division) set operations can be applied to intervals. ffl More importantly, with interval computing, many otherwise very difficult problems have been successfully solved [5, 15, 16, 14, 3]. 1 We use boldface letters and capital letters to denote interval quantities and vectors, respectively. For an interval variable, say x, x and x denote the greatest lower bound and the least upper bound, respectively. 1.3 Why INTBLAS The INTBLAS must rest on an underlying foundation of ....

R. Baker Kearfott. Treating non-smooth functions as smooth functions in global optimization and nonlinear systems solvers. In Gotz Alefeld, Andreas Frommer, and Bruno Lang, editors, Scientific Computing and Validated Numerics, Mathematical Research, pages 160--172. Akademie Verlag, Berlin, 1995.

.... available from Kearfott [6] Hansen [2] Neumaier [8] Special algorithms developed in GlobSol handle non smooth problems such as l 1 and l 1 optimizations with 3 the same techniques as smooth problems, and under certain conditions the interval Newton method converges linearly (See Kearfott [4] [5], Ratz [10] In practice, the interval Newton procedure can also be combined with an interval branch and bound technique, so that roots of g( 0 that cannot be the global minimum need not be found. The solution algorithm is applied to a sequence of intervals, beginning with some initial ....

R. B. Kearfott. Treating non-smooth functions as smooth functions in global optimization and nonlinear systems solvers. In G. Alefeld, A. Frommer, and B. Lang, editors, Scientic Computing and Validated Numerics, Mathematical Research, volume 90, pages 160-172, Berlin, 1996. Akademie Verlag. 9

.... to obtain S ] f; x; x) Interval Newton iteration with slope intervals can be e ective in global optimization and nonlinear systems solvers, especially when the derivatives of the objective function f have jump discontinuities, such as when f contains terms involving k k or max, 6] [7], 16] 20] Lemma 2.2. Let f : R R, and let x be an interval vector containing x. Then the limiting slope interval is given by lim w(x) 0 S ] f; x; x) lim inf x x f(x) f( x) x x ; lim sup x x f(x) f( x) x x = m; M ] Proof. By de nition, S ] f; x; x) is the ....

R. B. Kearfott. Treating non-smooth functions as smooth functions in global optimization and nonlinear systems solvers. In G. Alefeld, A. Frommer, and B. Lang, editors, Scientic Computing and Validated Numerics, Mathematical Research, volume 90, pages 160-172, Berlin, 1996. Akademie Verlag.

....X) if x q [ x q 0; S (d) jx q j; x q ; x q )S(x q ; X; X) otherwise, where S (d) jxj; x; x) h(x) h(x) with h(x) jxj Gammaj xj x Gamma x for x 62 x; Gamma1; 1] otherwise. The third branch of Formula 13 is an application of a generalization of [24, Theorem 3. 4] see [15]. 2.3 Formulas for x p = maxfx q ; x r g For real values x q and x r , maxfx q ; x r g = x r Gamma x q ; x q ; x r ) but (x r Gamma x q ; x q ; x r ) overestimates the range of max for interval values x q and x r . Formulas appropriate for max follow. Formula 14 Floating point evaluation ....

R. B. Kearfott. Treating non-smooth functions as smooth functions in global optimization and nonlinear systems solvers. In G. Alefeld and A. Frommer, editors, Scientific Computing and Validated Numerics, Mathematical Research, Berlin, 1995. Akademie Verlag.

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