GSLM Operations Research II Fall 13/14

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Deirdre Griffith
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1 GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are conssted of separable functons.e. f(x) = n f ( x ) and n g ( x ) b j = m; j j and all x are non-negatve varables bounded above.e. x for some = n. The technque separable programmng bascally replaces all separable functons n objectves and constrants by pecewse lnear functons. Defnton 6.. A convex program s an NLP that mnmzes a convex functon or maxmzes a concave functon over a convex set. Fact 6.. Any (contnuous) convex functon can be approxmated to any degree of accuracy by a pecewse lnear convex functon. From Fact 6. when f j and g j are convex functons they can be approxmated to any degree of accuracy by pecewse lnear functons. Eventually such an NLP of pecewse convex functons can be represented by a lnear program (LP). Thus effectvely a separable convex program can be approxmated by a sequence of LPs to any degree of accuracy. Note also that when f j and g j are convex an local mnmum s n fact a global mnmum. Example 6.. NLP s: mn x x x s.t. x + x 5 x + x 9 x x. The above s a separable program wth f (x ) = O A 4 B C

2 GSLM 58 Operatons Research II Fall /4 x x and f (x ) = -x. f s lnear. The problem s easy to solve f f.e. x s approxmated by a lnear functon. From the constrants x 4.5. Let us approxmate the non-lnear functon y = x by a pecewse lnear functon. For smplcty we take a functon of three lnear peces wth break ponts at 4 and 4.5. The same procedure can be appled to any number of break ponts at any values. ponts O A B C x y There are two ways the - and the -forms to represent the functon n pecewse lnear form. Defnton 6.. (-form) For any pont wthn a lnear segment ts functonal value s the convex combnaton of the values of the two break ponts of the lnear segment. Let be the weght of break pont = O A B and C. x O A 4B 4.5C y O 4 A 6B.5C O A B C at most two adjacent take non - zero values. () () () (4) Now reformulate NLP nto NLP mn y x x s.t. x + x 5 x + x 9 () () () and (4) x x = O A B C.

3 GSLM 58 Operatons Research II Fall /4 Constrants (4) can be omtted f NLP s a convex program. Check that whenever (4) s volated by a set of the value of the correspondng y s above the three-pece lnear functon and hence the set of cannot be a mnmum pont. In general f the NLP s a non-convex program constrants (4) are needed though they are mplemented mplctly through the separable programmng extenson of the smplex method. The pvotng step of the separable programmng extenson smply ensures that at any tme at most two adjacent are n the bass. It can be shown that such a restrcted bass rule wll lead to the optmum soluton. The dea of separable program s applcable to non-convex programs. Of course n those cases the optmum soluton can be a local rather than global optmum. Example 6.. (Non-Convex problem) mn f(x) s.t. x. f(x) 6 f(x) s a pecewse lnear functon (whch s not lnear by tself). It s obvous that mnmum s x * = wth f(x * ) =. Let us fnd ths by the dea of separable programmng. x Let x = + A + B and y = + A + 6 B ; and add n the constrant + A + B =. We take the specal attenton of (4) that at most two s can be postve at any tme. The problem becomes mn + A + 6 B s.t. + A + B + A + B + A + B = A B and at most two adjacent s can be postve at the same tme. After addng n slack varable s surplus varable u and two artfcal varables a and a the problem becomes mn + A + 6 B + Ma + Ma

4 GSLM 58 Operatons Research II Fall /4 s.t. + A + B + s = + A + B u + a = + A + B + a = A B s a a and at most two adjacent s can be postve at the same tme. A B s u a a RHS 6 M M s a - a A B s u a a RHS M M 64M M M s a - a A B s u a a RHS 5M 8 M 64M M s B a Supposedly s the most negatve. However snce B s n the bass only A among the s s qualfed to be n the bass. A B s u a a RHS 8 M 8M M M M M s A a 4

5 GSLM 58 Operatons Research II Fall /4 A B s u a a RHS 8+M +M M -+M s - - a A B s u a a RHS 9 5 M s - u Defnton 6.4. (-form) Instead of takng weghted value of break ponts we can add up the contrbuton from each lnear segment. Let be the proporton of the th segment taken =. there exsts j x y 4 such that for j and for j. (5) (6) (7) (8) As for the -form the -form may also gve a local optmum f the orgnal program s non-convex. When the orgnal program s convex constrant (8) s not necessary. Remark 6.. To get more accurate result the pecewse lnear approxmaton of f can be refned wth more lnear segments. There are studes on segment refnement to get the best trade off between accuracy and computatonal effort. Remark 6.. It s possble to approxmate constrants by smlar procedure. Remark 6.. A product term x x can be transformed to separable form. Let s = (x +x )/ s and s = (x -x )/. Then x x = s. 5

The Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique

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