Outline What is non-linear programming? Math 408A: Non-Linear Optimization Introduction Professor James Burke Math Dept, University of Washington Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Outline What is non-linear programming? What is non-linear programming? Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I The function to be minimized or maximized is called the objective function. I The set of alternatives is called the constraint region (or feasible region). I In this course, the feasible region is always taken to be a n subset of R (real n-dimensional space) and the objective n function is a function from R to R. Outline What is non-linear programming? What is optimization? A mathematical optimization problem is one in which a given function is either maximized or minimized relative to a given set of alternatives. Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I The set of alternatives is called the constraint region (or feasible region). I In this course, the feasible region is always taken to be a n subset of R (real n-dimensional space) and the objective n function is a function from R to R. Outline What is non-linear programming? What is optimization? A mathematical optimization problem is one in which a given function is either maximized or minimized relative to a given set of alternatives. I The function to be minimized or maximized is called the objective function. Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I In this course, the feasible region is always taken to be a n subset of R (real n-dimensional space) and the objective n function is a function from R to R. Outline What is non-linear programming? What is optimization? A mathematical optimization problem is one in which a given function is either maximized or minimized relative to a given set of alternatives. I The function to be minimized or maximized is called the objective function. I The set of alternatives is called the constraint region (or feasible region). Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Outline What is non-linear programming? What is optimization? A mathematical optimization problem is one in which a given function is either maximized or minimized relative to a given set of alternatives. I The function to be minimized or maximized is called the objective function. I The set of alternatives is called the constraint region (or feasible region). I In this course, the feasible region is always taken to be a n subset of R (real n-dimensional space) and the objective n function is a function from R to R. Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization X is the variable space (or decision space), f0 : X ! R [ {±∞} is the objective function, and Ω ⊂ X is the constraint region. Outline What is non-linear programming? What is optimization? P : minimize f0(x) x2X subject to x 2 Ω: Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization f0 : X ! R [ {±∞} is the objective function, and Ω ⊂ X is the constraint region. Outline What is non-linear programming? What is optimization? P : minimize f0(x) x2X subject to x 2 Ω: X is the variable space (or decision space), Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Ω ⊂ X is the constraint region. Outline What is non-linear programming? What is optimization? P : minimize f0(x) x2X subject to x 2 Ω: X is the variable space (or decision space), f0 : X ! R [ {±∞} is the objective function, and Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Outline What is non-linear programming? What is optimization? P : minimize f0(x) x2X subject to x 2 Ω: X is the variable space (or decision space), f0 : X ! R [ {±∞} is the objective function, and Ω ⊂ X is the constraint region. Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z I unconstrained: Ω = X I constrained: Ω 6= X I Variable Type: I Constraint Type: Outline What is non-linear programming? What is optimization? Problem Categories: Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I unconstrained: Ω = X I constrained: Ω 6= X n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z I Constraint Type: Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I unconstrained: Ω = X I constrained: Ω 6= X n I discrete variable: X = Z s t I mixed variable: X = R × Z I Constraint Type: Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I unconstrained: Ω = X I constrained: Ω 6= X s t I mixed variable: X = R × Z I Constraint Type: Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R n I discrete variable: X = Z Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I unconstrained: Ω = X I constrained: Ω 6= X I Constraint Type: Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I unconstrained: Ω = X I constrained: Ω 6= X Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z I Constraint Type: Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization I constrained: Ω 6= X Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z I Constraint Type: I unconstrained: Ω = X Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Outline What is non-linear programming? What is optimization? Problem Categories: I Variable Type: n I continuous variable: X = R n I discrete variable: X = Z s t I mixed variable: X = R × Z I Constraint Type: I unconstrained: Ω = X I constrained: Ω 6= X Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization Convex Programming: f0 a convex function, Ω a convex set n Definition: Ω ⊂ R is convex if for every x; y 2 Ω one has [x; y] ⊂ Ω where [x; y] is the line segment connecting x and y: [x; y] = fλx + (1 − λ)y : 0 ≤ λ ≤ 1g: n Definition: f : R ! R [ {±∞} is convex if n+1 epi(f ) = f(x; µ): f (x) ≤ µg is a convex set in R . In particular, f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all 0 ≤ λ ≤ 1 and points x; y for which not both f (x) and f (y) are infinite. Outline What is non-linear programming? Problem Types Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization f0 a convex function, Ω a convex set n Definition: Ω ⊂ R is convex if for every x; y 2 Ω one has [x; y] ⊂ Ω where [x; y] is the line segment connecting x and y: [x; y] = fλx + (1 − λ)y : 0 ≤ λ ≤ 1g: n Definition: f : R ! R [ {±∞} is convex if n+1 epi(f ) = f(x; µ): f (x) ≤ µg is a convex set in R . In particular, f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all 0 ≤ λ ≤ 1 and points x; y for which not both f (x) and f (y) are infinite. Outline What is non-linear programming? Problem Types Convex Programming: Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization n Definition: Ω ⊂ R is convex if for every x; y 2 Ω one has [x; y] ⊂ Ω where [x; y] is the line segment connecting x and y: [x; y] = fλx + (1 − λ)y : 0 ≤ λ ≤ 1g: n Definition: f : R ! R [ {±∞} is convex if n+1 epi(f ) = f(x; µ): f (x) ≤ µg is a convex set in R . In particular, f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all 0 ≤ λ ≤ 1 and points x; y for which not both f (x) and f (y) are infinite. Outline What is non-linear programming? Problem Types Convex Programming: f0 a convex function, Ω a convex set Introduction Professor James Burke Math Dept, University of Washington Math 408A: Non-Linear Optimization n Definition: f : R ! R [ {±∞} is convex if n+1 epi(f ) = f(x; µ): f (x) ≤ µg is a convex set in R . In particular, f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all 0 ≤ λ ≤ 1 and points x; y for which not both f (x) and f (y) are infinite.