R. Fourer and D.M. Gay, Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization Robert Fourer Department of Industrial Engineering & Management Sciences Northwestern University David M. Gay AMPL Optimization LLC 20th Biennial Conference on Numerical Analysis University of Dundee, Scotland — 24-27 June 2003 1 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 Abstract The idea of a modeling language is to describe mathematical problems in a symbolic form that is familiar to people, but that can be processed by computer systems. In particular the concept of an algebraic modeling language, based on objective and constraint expressions in terms of decision variables, has proved to be valuable for a broad range of optimization and related problems. One modeling language can work with numerous solvers, each of which implements one or more optimization algorithms. The separation of model specification from solver execution is thus a key tenet of modeling language design. Nevertheless, several issues in numerical analysis that are critical to solvers are also important in implementations of modeling languages. So-called presolve procedures, which tighten bounds with the aim of eliminating some variables and constraints, are numerical algorithms that require carefully chosen tolerances and can benefit from directed roundings. Correctly rounded binary-decimal conversion is valuable in portably conveying problem instances and in debugging. Further rounding options offer tradeoffs between accuracy, convenience, and readability in displaying numerical data. Modeling languages can also strongly influence the development of solvers. Most notably, for smooth nonlinear optimization, the ability to provide numerically computed, exact first and second derivatives has made modeling languages a valuable tool in solver development. The generality of modeling languages has also encouraged the development of more general solvers, such as for optimization problems with equilibrium constraints. This presentation draws from our experience in developing the AMPL modeling language to provide examples in all of the above areas. We conclude by describing possibilities for future work that would have a significant numerical aspect. 2 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, June 24-27, 2003
R. Fourer and D.M. Gay, Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization Outline Rounding and conversion Displayed vs. actual values Correctly rounded conversions Presolving Fixed variables, redundant constraints Infeasible constraints Influence on solvers Second derivatives ↔ IP solvers Complementarity problems ↔ MPECs Future influences Quadratic expressions Matrix functions and constraints Nonlinear expressions as input to solvers 3 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 A Brief Introduction to AMPL: The McDonald’s Diet Problem Foods: Nutrients: QP Quarter Pounder Prot Protein FR Fries, small Iron Iron MD McLean Deluxe VitA Vitamin A SM Sausage McMuffin Cals Calories BM Big Mac VitC Vitamin C 1M 1% Lowfat Milk Carb Carbohydrates FF Filet-O-Fish Calc Calcium OJ Orange Juice MC McGrilled Chicken 4 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, June 24-27, 2003
R. Fourer and D.M. Gay, Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization McDonald’s Diet Problem Data QP MD BM FF MC FR SM 1M OJ Cost 1.8 2.2 1.8 1.4 2.3 0.8 1.3 0.6 0.7 Need: Protein 55 28 24 25 14 31 3 15 9 1 Vitamin A 15 15 6 2 8 0 4 10 2 100 Vitamin C 6 10 2 0 15 15 0 4 120 100 Calcium 100 30 20 25 15 15 0 20 30 2 Iron 20 20 20 10 8 2 15 0 2 100 Calories 510 370 500 370 400 220 345 110 80 2000 Carbo 34 35 42 38 42 26 27 12 20 350 5 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 Formulation: Too General Minimize cx Subject to Ax = b x ≥ 0 6 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, June 24-27, 2003
R. Fourer and D.M. Gay, Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization Formulation: Too Specific Minimize 1.84 x QP + 2.19 x MD + 1.84 x BM + 1.44 x FF + 2.29 x MC + 0.77 x FR + 1.29 x SM + 0.60 x 1M + 0.72 x OJ Subject to 28 x QP + 24 x MD + 25 x BM + 14 x FF + 31 x MC + 3 x FR + 15 x SM + 9 x 1M + 1 x OJ ≥ 55 15 x QP + 15 x MD + 6 x BM + 2 x FF + 8 x MC + 0 x FR + 4 x SM + 10 x 1M + 2 x OJ ≥ 100 6 x QP + 10 x MD + 2 x BM + 0 x FF + 15 x MC + 15 x FR + 0 x SM + 4 x 1M + 120 x OJ ≥ 100 30 x QP + 20 x MD + 25 x BM + 15 x FF + 15 x MC + 0 x FR + 20 x SM + 30 x 1M + 2 x OJ ≥ 100 20 x QP + 20 x MD + 20 x BM + 10 x FF + 8 x MC + 2 x FR + 15 x SM + 0 x 1M + 2 x OJ ≥ 100 510 x QP + 370 x MD + 500 x BM + 370 x FF + 400 x MC + 220 x FR + 345 x SM + 110 x 1M + 80 x OJ ≥ 2000 34 x QP + 35 x MD + 42 x BM + 38 x FF + 42 x MC + 26 x FR + 27 x SM + 12 x 1M + 20 x OJ ≥ 350 7 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 Formulation: Algebraic Model Given F , a set of foods N , a set of nutrients and a ij ≥ 0, the units of nutrient i in one serving of food j , for each i ∈ N and j ∈ F b i > 0, the units of nutrient i required, for each i ∈ N c j > 0, the cost per serving of food j , for each j ∈ F x j ≥ 0, the number of servings of food j to be purchased, for each j ∈ F Define Σ j ∈ F c j x j Minimize Σ j ∈ F a ij x j ≥ b i , for each i ∈ N Subject to 8 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, June 24-27, 2003
R. Fourer and D.M. Gay, Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization Algebraic Model in AMPL set NUTR; # nutrients set FOOD; # foods param amt {NUTR,FOOD} >= 0; # amount of nutrient in each food param nutrLow {NUTR} >= 0; # lower bound on nutrients in diet param cost {FOOD} >= 0; # cost of foods var Buy {FOOD} >= 0 integer; # amounts of foods to be purchased minimize TotalCost: sum {j in FOOD} cost[j] * Buy[j]; subject to Need {i in NUTR}: sum {j in FOOD} amt[i,j] * Buy[j] >= nutrLow[i]; 9 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 Data for the AMPL Model param: FOOD: cost := "Quarter Pounder" 1.84 "Fries, small" .77 "McLean Deluxe" 2.19 "Sausage McMuffin" 1.29 "Big Mac" 1.84 "1% Lowfat Milk" .60 "Filet-O-Fish" 1.44 "Orange Juice" .72 "McGrilled Chicken" 2.29 ; param: NUTR: nutrLow := Prot 55 VitA 100 VitC 100 Calc 100 Iron 100 Cals 2000 Carb 350 ; param amt (tr): Cals Carb Prot VitA VitC Calc Iron := "Quarter Pounder" 510 34 28 15 6 30 20 "McLean Deluxe" 370 35 24 15 10 20 20 "Big Mac" 500 42 25 6 2 25 20 "Filet-O-Fish" 370 38 14 2 0 15 10 "McGrilled Chicken" 400 42 31 8 15 15 8 "Fries, small" 220 26 3 0 15 0 2 "Sausage McMuffin" 345 27 15 4 0 20 15 "1% Lowfat Milk" 110 12 9 10 4 30 0 "Orange Juice" 80 20 1 2 120 2 2 ; 10 Robert Fourer & David M. Gay, 20th Biennial Conference on Numerical Analysis, Dundee, Scotland — 24-27 June 2003 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, June 24-27, 2003
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