A Practical Guide to Mixed Integer Nonlinear Programming (MINLP) Sven Leyffer Jeff Linderoth MCS Division ISE Department Argonne National Lab Lehigh University [email protected] [email protected] SIAM Conference on Optimization Stockholm, Sweden May 15, 2005 Leyffer & Linderoth MINLP NLP MINLP + =6 New Math MI Leyffer & Linderoth MINLP MINLP =6 New Math MI NLP + Leyffer & Linderoth MINLP New Math MI NLP MINLP + =6 Leyffer & Linderoth MINLP 2. Classical Solution Methods 3. Modern Developments in MINLP 4. Implementation and Software MINLP Short Course Overview 1. Introduction, Applications, and Formulations Leyffer & Linderoth MINLP 3. Modern Developments in MINLP 4. Implementation and Software MINLP Short Course Overview 1. Introduction, Applications, and Formulations 2. Classical Solution Methods Leyffer & Linderoth MINLP 4. Implementation and Software MINLP Short Course Overview 1. Introduction, Applications, and Formulations 2. Classical Solution Methods 3. Modern Developments in MINLP Leyffer & Linderoth MINLP MINLP Short Course Overview 1. Introduction, Applications, and Formulations 2. Classical Solution Methods 3. Modern Developments in MINLP 4. Implementation and Software Leyffer & Linderoth MINLP Motivation Examples Tricks Part I Introduction, Applications, and Formulations Leyffer & Linderoth MINLP • f, c smooth (convex) functions • X, Y polyhedral sets, e.g. Y = {y ∈ [0, 1]p | Ay ≤ b} • y ∈ Y integer ⇒ hard problem • f, c not convex ⇒ very hard problem Motivation What Examples How Tricks Why? The Problem of the Day Mixed Integer Nonlinear Program (MINLP) minimize f(x, y) x,y subject to c(x, y) ≤ 0 x ∈ X, y ∈ Y integer Leyffer & Linderoth MINLP • y ∈ Y integer ⇒ hard problem • f, c not convex ⇒ very hard problem Motivation What Examples How Tricks Why? The Problem of the Day Mixed Integer Nonlinear Program (MINLP) minimize f(x, y) x,y subject to c(x, y) ≤ 0 x ∈ X, y ∈ Y integer • f, c smooth (convex) functions • X, Y polyhedral sets, e.g. Y = {y ∈ [0, 1]p | Ay ≤ b} Leyffer & Linderoth MINLP • f, c not convex ⇒ very hard problem Motivation What Examples How Tricks Why? The Problem of the Day Mixed Integer Nonlinear Program (MINLP) minimize f(x, y) x,y subject to c(x, y) ≤ 0 x ∈ X, y ∈ Y integer • f, c smooth (convex) functions • X, Y polyhedral sets, e.g. Y = {y ∈ [0, 1]p | Ay ≤ b} • y ∈ Y integer ⇒ hard problem Leyffer & Linderoth MINLP Motivation What Examples How Tricks Why? The Problem of the Day Mixed Integer Nonlinear Program (MINLP) minimize f(x, y) x,y subject to c(x, y) ≤ 0 x ∈ X, y ∈ Y integer • f, c smooth (convex) functions • X, Y polyhedral sets, e.g. Y = {y ∈ [0, 1]p | Ay ≤ b} • y ∈ Y integer ⇒ hard problem • f, c not convex ⇒ very hard problem Leyffer & Linderoth MINLP • Equilibrium • Enthalpy • Physical Processes and Properties • Abstract Measures • Economies of But we all know the world is Scale nonlinear! • Covariance • Utility of decisions Motivation What Examples How Tricks Why? Why the N? An anecdote: July, 1948. A young and frightened George Dantzig, presents his newfangled “linear programming” to a meeting of the Econometric Society of Wisconsin, attended by distinguished scientists like Hotelling, Koopmans, and Von Neumann. Following the lecture, Hotellinga pronounced to the audience: ain Dantzig’s words “a huge whale of a man” Leyffer & Linderoth MINLP • Equilibrium • Enthalpy • Physical Processes and Properties • Abstract Measures • Economies of Scale • Covariance • Utility of decisions Motivation What Examples How Tricks Why? Why the N? An anecdote: July, 1948. A young and frightened George Dantzig, presents his newfangled “linear programming” to a meeting of the Econometric Society of Wisconsin, attended by distinguished scientists like Hotelling, Koopmans, and Von Neumann. Following the lecture, Hotellinga pronounced to the audience: But we all know the world is nonlinear! ain Dantzig’s words “a huge whale of a man” Leyffer & Linderoth MINLP • Equilibrium • Enthalpy • Physical Processes and Properties • Abstract Measures • Economies of Scale • Covariance • Utility of decisions Motivation What Examples How Tricks Why? Why the N? An anecdote: July, 1948. A young and frightened George Dantzig, presents his The world is indeed newfangled “linear programming” to a nonlinear meeting of the Econometric Society of Wisconsin, attended by distinguished scientists like Hotelling, Koopmans, and Von Neumann. Following the lecture, Hotellinga pronounced to the audience: But we all know the world is nonlinear! ain Dantzig’s words “a huge whale of a man” Leyffer & Linderoth MINLP • Abstract Measures • Economies of Scale • Covariance • Utility of decisions Motivation What Examples How Tricks Why? Why the N? An anecdote: July, 1948. A young and frightened George Dantzig, presents his The world is indeed newfangled “linear programming” to a nonlinear meeting of the Econometric Society of Wisconsin, attended by distinguished • Physical Processes scientists like Hotelling, Koopmans, and and Properties Von Neumann. Following the lecture, • Equilibrium Hotellinga pronounced to the audience: • Enthalpy But we all know the world is nonlinear! ain Dantzig’s words “a huge whale of a man” Leyffer & Linderoth MINLP Motivation What Examples How Tricks Why? Why the N? An anecdote: July, 1948. A young and frightened George Dantzig, presents his The world is indeed newfangled “linear programming” to a nonlinear meeting of the Econometric Society of Wisconsin, attended by distinguished • Physical Processes scientists like Hotelling, Koopmans, and and Properties Von Neumann. Following the lecture, • Equilibrium Hotellinga pronounced to the audience: • Enthalpy • Abstract Measures But we all know the world is nonlinear! • Economies of Scale • Covariance ain Dantzig’s words “a huge whale of a • man” Utility of decisions Leyffer & Linderoth MINLP • If the variable is associated with a physical entity that is indivisible, then it must be integer 1. Number of aircraft carriers to to produce. Gomory’s Initial Motivation 2. Yearly number of trees to harvest in Norrland Motivation What Examples How Tricks Why? Why the MI? • We can use 0-1 (binary) variables for a variety of purposes • Modeling yes/no decisions • Enforcing disjunctions • Enforcing logical conditions • Modeling fixed costs • Modeling piecewise linear functions Leyffer & Linderoth MINLP Motivation What Examples How Tricks Why? Why the MI? • We can use 0-1 (binary) variables for a variety of purposes • Modeling yes/no decisions • Enforcing disjunctions • Enforcing logical conditions • Modeling fixed costs • Modeling piecewise linear functions • If the variable is associated with a physical entity that is indivisible, then it must be integer 1. Number of aircraft carriers to to produce. Gomory’s Initial Motivation 2. Yearly number of trees to harvest in Norrland Leyffer & Linderoth MINLP 1. Convince the user that he or she does not wish to solve a mixed integer nonlinear programming problem at all! 2. Otherwise, solve the continuous relaxation (NLP ) and round off the minimizer to the nearest integer. • Sometimes a continuous approximation to the discrete (integer) decision is accurate enough for practical purposes. • Yearly tree harvest in Norrland • For 0 − 1 problems, or those in which the |y| is “small”, the continuous approximation to the discrete decision is not accurate enough for practical purposes. • Conclusion: MINLP methods must be studied! Motivation What Examples How Tricks Why? A Popular MINLP Method Dantzig’s Two-Phase Method for MINLP Adapted by Leyffer and Linderoth Leyffer & Linderoth MINLP 2. Otherwise, solve the continuous relaxation (NLP ) and round off the minimizer to the nearest integer. • Sometimes a continuous approximation to the discrete (integer) decision is accurate enough for practical purposes. • Yearly tree harvest in Norrland • For 0 − 1 problems, or those in which the |y| is “small”, the continuous approximation to the discrete decision is not accurate enough for practical purposes. • Conclusion: MINLP methods must be studied! Motivation What Examples How Tricks Why? A Popular MINLP Method Dantzig’s Two-Phase Method for MINLP Adapted by Leyffer and Linderoth 1. Convince the user that he or she does not wish to solve a mixed integer nonlinear programming problem at all! Leyffer & Linderoth MINLP • Sometimes a continuous approximation to the discrete (integer) decision is accurate enough for practical purposes. • Yearly tree harvest in Norrland • For 0 − 1 problems, or those in which the |y| is “small”, the continuous approximation to the discrete decision is not accurate enough for practical purposes. • Conclusion: MINLP methods must be studied! Motivation What Examples How Tricks Why? A Popular MINLP Method Dantzig’s Two-Phase Method for MINLP Adapted by Leyffer and Linderoth 1. Convince the user that he or she does not wish to solve a mixed integer nonlinear programming problem at all! 2. Otherwise, solve the continuous relaxation (NLP ) and round off the minimizer to the nearest integer. Leyffer & Linderoth MINLP Motivation What Examples How Tricks Why? A Popular MINLP Method Dantzig’s Two-Phase Method for MINLP Adapted by Leyffer and Linderoth 1. Convince the user that he or she does not wish to solve a mixed integer nonlinear programming problem at all! 2. Otherwise, solve the continuous relaxation (NLP ) and round off the minimizer to the nearest integer. • Sometimes a continuous approximation to the discrete (integer) decision is accurate enough for practical purposes. • Yearly tree harvest in Norrland • For 0 − 1 problems, or those in which