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Optimization

Optimization. Topic: Fundamental concepts and algorithmic methods for solving linear and integer linear programs : Simplex method LP duality Ellipsoid method … Prerequisites : Basic math and theory courses Credit : 4+2 hours => 9 CP; written end of term exam

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Optimization

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  1. Optimization • Topic: Fundamental concepts and algorithmic methods for solving linear and integer linear programs: • Simplex method • LP duality • Ellipsoid method • … • Prerequisites: Basic math and theory courses • Credit: 4+2 hours => 9 CP; written end oftermexam • Lecturers: Dr. Reto Spöhel, PD Dr. Rob van Stee • Tutors: Karl Bringmann, Ruben Becker • Lectures: Tue 10 ̶12 & Thu 12 ̶14 in E1.4 (MPI-INF), room 0.24Exercises: Wed 10 ̶12 / 14 ̶16

  2. Optimization Given: • a targetfunctionf(x1, …, xn) of decision variables x1, …, xn2 R • e.g. f(x1, …, xn) = x1 + 5x3 – 2x7 • orf(x1, …, xn) = x1¢ (x2 + x3) / x8 • a set of constraintsthe variables x1, …, xnneed to satisfy • e.g. x1 ¸ 0andx2¢x3·x4and x4 + x5 = x6 andx1, x3, x52Z Goal: • find x1, …, xn2 R minimizing fsubject to thegivenconstraints Geometricviewpoint: Theset of points (x1, …, xn) 2 Rnthatsatisfy all constraintsiscalledthefeasibleregion of theoptimizationproblem. I.e., find (x1, …, xn) insidethefeasibleregionthatminimizes f among all such points

  3. Optimization • Importantspecialcases: • convexoptimization: Boththefeasibleregion and thetargetfunctionareconvex  anylocalminimumis a global minimum (!) • ¶Semi-Definite Programming: constraintsarequadratic and semidefinite; targetfunctionislinear • ¶ QuadraticProgramming: constraintsarelinear, targetfunctionisquadratic and semidefinite • ¶ Linear Programming: constraintsarelinear, targetfunctionislinear • Integer [Linear] Programming: Linear programmingwith additional constraintthatsomeor all decision variable mustbeintegral (oreven2 {0,1}) Thiscourseismostlyabout LP and IP!

  4. Linear Programming – A First Example [Exampletakenfromlecturenotesby Carl W. Lee, U. of Kentucky] • A companymanufacturesgadgets and gewgaws • One kg of gadgets • requires1 hourof work, 1 unitof wood, and 2 unitsof metal • yields a netprofit of $5 • One kg of gewgaws • requires2 hoursof work, 1 unitof wood, and 1 unitof metal • yields a netprofit of $4 • Thecompany has 120 hoursof work, 70 unitsof wood, and 100 unitsof metal available • Whatshoulditproducefromtheseresources to maximizeitsprofit?

  5. Linear Programming – A First Example • One kg of gadgets • requires1 hourof work, 1 unitof wood, and 2 unitsof metal • yields a netprofit of $5 • One kg of gewgaws • requires2 hoursof work, 1 unitof wood, and 1 unitof metal • yields a netprofit of $4 • Thecompany has 120 hoursof work, 70 unitsof wood, and100 unitsof metal available maximizeprofit workhoursconstraint x1 := amount of gadgets woodconstraint x2 := amount of gewgaws metal constraint

  6. Linear Programming – A First Example maximizeprofit workhoursconstraint x2 = amount of gewgaws woodconstraint metal constraint x2 = 100 feasibleregion all pointswithprofit 5x1 + 4x2 = 100 ? x2 = 70 all pointswithprofit 5x1 + 4x2 = 200 x2 = 60 all pointswithprofit 5x1 + 4x2 = 310 theonly point withprofit 310 insidethefeasibleregion --- theoptimum! x2 = 40 x2 = 25 x1 = amount of gadgets x1 = 30 x1 = 120 x1 = 20 x1 = 50 x1 = 70

  7. Observations / Intuitions • Itseemsthat in Linear Programming • thefeasibleregionisalways a convexpolygon/polytope • theoptimumisalwaysattained at one of thecorners of thispolygon/polytope • These intuitionsaremoreorlesstrue! • In thecomingweeks, we will makethemprecise, and exploitthem to derivealgorithmsforsolving linear programs. • First and foremost: The simplexmethod

  8. LPs in Theoretical CS • Manyother CS problemscanbecast as LPs. • Example: MAXFLOW canbewritten as thefollowing LP: • This LP has 2|E| + |V|-2 manyconstraintsand |E| manydecision variables. • LPs canbesolved in time polynomial in theinputsize! • Butthealgorithmmostcommonlyusedforsolvingthem in practice (simplexalgorithm) isnot a polynomialalgorithm!

  9. Books • Main referenceforthecourse: Introduction to Linear Optimizationby Dimitris Bertsimas and John N. Tsitsiklis • ratherexpensive to buy • thelibrary has ca. 20 copies • PDF of Chapters 1-5: google „leenstougie linear programming“ • Secondaryreference: CombinatorialOptimization: Algorithms and Complexityby Christos H. Papadimitriou and Kenneth Steiglitz • veryinexpensive (ca. €15 on amazon.de)  • old (1982) but covers most of thebasics classic!

  10. Exercises • Exercisesessions start in week 3 of thesemester • Twogroups: • Wed 10-12 in E1.7 (clusterbuilding), room 001Tutor: Ruben • Wed 14-16 in E 1.4 (thisbuilding), room 023Tutor: Karl • Courseregistration and assignment to groupsnextweek! • Exercisesheets • will beputonlineTuesdayeveningeachweek • shouldbehanded in thefollowingTuesday in thelecture • Youneed to achieve 50% of all availablepoints in thefirstand in the second half of theterm to beadmitted to theexam.

  11. That‘sitfortoday. On Thursdaywe will start withthedefinitions and theorems.

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