3rd Int’l Conf. on Numerical Analysis and Optimization Sultan Qaboos University, Muscat, Oman, 5–9 Jan. 2014 The AMPL Modeling Language — an Aid to Formulating and Solving Optimization Problems David M. Gay AMPL Optimization, Inc. [email protected] http://www.ampl.com 1 AMPL summary AMPL: a language for mathematical programming problems, e.g., minimize f (x) s.t. ℓ ≤ c(x) ≤ u, Rn Rn Rm with x ∈ and c : → given algebraically and some x discrete. i 2 AMPL goals • Easy transcription from math (avoid mistakes) •• Explicit indexing (no hidden magic) • Declare before use (one-pass reading) • Separate model, data, commands (orthogonality) • Separate solvers (open solver interface) • Update entities as needed (lazy eval.) ... • Builtin math. prog. stuff (presolve, red. costs, ) • Aim for large scale nonlinear (sparsity, generality) 3 AMPL history • Early 1980’s: little languages at Bell Labs • 1983: Fourer’s TOMS paper, ML’s vs MG’s • 1984: Karmarkar hoopla begins • 1985–6: Fourer sabbatical at Bell Labs • 1987: AMPL CSTR (Fourer, Gay, Kernighan) • Late 1980’s: reimplementation; extensions begin • 1993: 1st edition of AMPL book • 2001: collapse of Bell Labs • 2003: 2nd edition of AMPL book; dmg → Sandia • 2010: dmg → AMPL Optimization 4 Simple declarations, commands ampl: param p; ampl: param q = p + 10; ampl: data; param p := 2.5; ampl data: display p, q; p = 2.5 q = 12.5 ampl: let p := 17; display p, q; p = 17 q = 27 5 Simple sets ampl: set A; set B; ampl: set C = p .. q; ampl: display A; Error executing "display" command: no data for set A ampl: data; set A := a b c; set B := c d; ampl data: display A, B, C; set A := a b c; set B := c d; set C := 17 18 19 20 21 22 23 24 25 26 27; 6 Simple set operations ampl: display A intersect B, A union B; set A inter B := c; set A union B := a b c d; ampl: display A diff B, A symdiff B; set A diff B := a b; set A symdiff B := a b d; 7 Iterated expressions, declarations ampl: print sum {i in 1..4} i; 10 ampl: print prod {i in 1..4} i; 24 ampl: param fac{ i in 1..9 } ampl? = if i == 1 then 1 else i*fac[i-1]; ampl: print max{i in 1..9} ampl? abs(fac[i] - prod{j in 2..i} j); 0 8 More iterated commands ampl: display fac, {i in 1..9} prod{j in 2..i} j; : fac prod{j in 2 .. i} j := 1 1 1 2 2 2 3 6 6 4 24 24 5 120 120 6 720 720 7 5040 5040 8 40320 40320 9 362880 362880 ; 9 Example model: diet.mod set NUTR; set FOOD; param cost {FOOD} > 0; param f_min {FOOD} >= 0; param f_max {j in FOOD} >= f_min[j]; param n_min {NUTR} >= 0; param n_max {i in NUTR} >= n_min[i]; param amt {NUTR,FOOD} >= 0; var Buy {j in FOOD} >= f_min[j], <= f_max[j]; minimize Total_Cost: sum {j in FOOD} cost[j] * Buy[j]; subject to Diet {i in NUTR}: n_min[i] <= sum {j in FOOD} amt[i,j] * Buy[j] <= n_max[i]; 10