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Connecting Calculus to Linear Programming Dimensional Optimization

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Connecting Calculus to Linear Programming Dimensional Optimization MIST 2017 Blais Optimization Linearity Higher- Connecting Calculus to Linear Programming Dimensional Optimization Higher- Dimensional Marcel Y. Blais, Ph.D. Linearity Worcester Polytechnic Institute Linear Programming Simplex Method Dept. of Mathematical Sciences Applications July 2017 Blais MIST 2017 We can connect calculus to real-world industrial problems. We'll explore the some basic principles of linear programming and some modern-day applications. Motivation MIST 2017 Blais Optimization Goal: To help students make connections between high school Linearity math and real world applications of mathematics. Higher- Dimensional Optimization Linear programming is based on a simple idea from Higher- calculus. Dimensional Linearity Linear Programming Simplex Method Applications Blais MIST 2017 We'll explore the some basic principles of linear programming and some modern-day applications. Motivation MIST 2017 Blais Optimization Goal: To help students make connections between high school Linearity math and real world applications of mathematics. Higher- Dimensional Optimization Linear programming is based on a simple idea from Higher- calculus. Dimensional Linearity We can connect calculus to real-world industrial problems. Linear Programming Simplex Method Applications Blais MIST 2017 Motivation MIST 2017 Blais Optimization Goal: To help students make connections between high school Linearity math and real world applications of mathematics. Higher- Dimensional Optimization Linear programming is based on a simple idea from Higher- calculus. Dimensional Linearity We can connect calculus to real-world industrial problems. Linear Programming We'll explore the some basic principles of linear Simplex Method programming and some modern-day applications. Applications Blais MIST 2017 Continuous Functions on Closed Intervals MIST 2017 Blais 5 Optimization 4 Linearity 3 Higher- Dimensional 2 Optimization y Higher- 1 Dimensional Linearity 0 Linear -1 Programming Simplex Method -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Applications a x b What can we say about a continuous function on a closed interval? Blais MIST 2017 Continuous Functions on Closed Intervals MIST 2017 5 Blais 4 Optimization 3 Linearity Higher- 2 Dimensional y Optimization 1 Higher- 0 Dimensional Linearity -1 Linear -2 Programming -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Simplex Method a x b Applications Theorem If f is continuous on [a; b] then f attains both an absolute minimum value and an absolute maximum value on [a; b]. Blais MIST 2017 Continuous Functions on Closed Intervals MIST 2017 5 Blais 4 Optimization 3 Linearity Higher- 2 Dimensional y Optimization 1 Higher- 0 Dimensional Linearity -1 Linear -2 Programming -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Simplex Method a x b Applications Theorem If f is continuous on [a; b] then f attains both an absolute minimum value and an absolute maximum value on [a; b]. Further these occur at critical points of f or endpoints of [a; b]. Blais MIST 2017 Linear Case MIST 2017 Theorem Blais If f is continuous linear on [a; b] then f attains both an Optimization absolute minimum value and an absolute maximum value on Linearity Higher- [a; b], and these occur at endpoints of [a; b]. Dimensional Optimization 35 Higher- Dimensional 30 Linearity Linear 25 Programming Simplex Method 20 Applications y 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 a x b Blais MIST 2017 Higher-Dimensional Optimization MIST 2017 Blais Optimization 0.2 Linearity 0.1 Higher- 0 Dimensional Optimization -0.1 z Higher- -0.2 Dimensional -0.3 Linearity -0.4 Linear 1 0.8 -0.5 0.6 Programming 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 Simplex Method 0.4 0.3 0.2 0.1 0 0 x Applications y What does this mean for a continuous function f (x; y) on a closed and bounded plane region R? Blais MIST 2017 Higher-Dimensional Optimization MIST 2017 Blais Optimization 0.2 Linearity 0.1 Higher- 0 Dimensional Optimization -0.1 z Higher- -0.2 Dimensional -0.3 Linearity -0.4 Linear 1 0.8 -0.5 0.6 Programming 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 Simplex Method 0.4 0.3 0.2 0.1 0 0 x y Applications Theorem If f (x; y) is continuous on closed and bounded plane region R then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R. Blais MIST 2017 Higher-Dimensional Optimization MIST 2017 Blais Optimization 0.2 Linearity 0.1 Higher- 0 Dimensional Optimization -0.1 z Higher- -0.2 Dimensional -0.3 Linearity -0.4 Linear 1 0.8 -0.5 0.6 Programming 1 0.9 0.8 0.4 0.7 0.6 Simplex Method Theorem 0.5 0.4 0.2 0.3 0.2 0.1 0 0 x Applications y If f (x1; x2;:::; xn) is continuous on closed and bounded region n R ⊂ R then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur either at critical points of f in the interior of R or on the boundary of R. Blais MIST 2017 Higher-Dimensional Linearity MIST 2017 5 4.5 Blais 4 3.5 Optimization 3 Linearity 2.5 z Higher- 2 Dimensional 1.5 Optimization 1 Higher- 0.5 Dimensional 0 1 Linearity 0.8 0.6 0.4 1 0.8 0.9 0.6 0.7 0.2 0.4 0.5 0.2 0.3 Linear y 0 0 0.1 Programming x Simplex Method Applications Theorem If f (x1; x2;:::; xn) is linear on closed and bounded region n R ⊂ R defined by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on the boundary of R. Blais MIST 2017 Higher-Dimensional Linearity MIST 2017 5 Blais 4.5 4 3.5 3 2.5 Optimization z 2 Linearity 1.5 1 Higher- 0.5 0 Dimensional 1 0.8 0.6 Optimization 0.4 0.9 1 0.7 0.8 0.2 0.6 0.4 0.5 0.2 0.3 y 0 0 0.1 Higher- x Dimensional Linearity Linear Programming Theorem Simplex Method If f (x ; x ;:::; x ) is linear on closed and bounded region Applications 1 2 n n R ⊂ R defined by a system of linear constraints then f attains both an absolute minimum value and an absolute maximum value on R. Further, these values occur on corner points of the boundary of R. Blais MIST 2017 Higher-Dimensional Linearity MIST 2017 Blais 1 0.9 Optimization 0.8 Linearity 0.7 0.6 Higher- 0.5 z Dimensional 0.4 Optimization 0.3 0.2 Higher- 0.1 Dimensional 0 1 Linearity 0.9 0.8 1 0.7 0.6 0.8 0.5 Linear 0.6 0.4 0.3 0.4 Programming 0.2 y 0.2 0.1 x Simplex Method 0 0 Applications Example: A possible feasible region f (x; y; z) = ax + by + cz subjuect to 5 linear constraints. A solution to minimize f over this region will occur at a corner. Blais MIST 2017 Linear Programming MIST 2017 Blais The Simplex Method was developed by George Dantzig in Optimization Linearity 1947. Higher- Dimensional \programming" synonymous with \optimization". Optimization Algorithm to traverse the corner points of the feasible Higher- Dimensional polyhedron for a linear programming problem to find an Linearity optimal feasible solution. Linear Programming Simplex Method Standard form for a linear programming problem: Applications min xT c such that Ax ≤ b and x ≥ 0: (1) n m m×n x; c 2 R ; b 2 R ; A 2 R - n variables and m constraints. Blais MIST 2017 Simplex Method MIST 2017 Blais min xT c such that Ax ≤ b and x ≥ 0. Optimization Linearity For each constraint (row i of A), add a new slack Higher- Dimensional variable yi . Optimization T x Higher- New system: min x c such that [A + I ] = b, Dimensional y Linearity x ≥ 0; y ≥ 0. Linear Programming Underdetermined (more variables than equations). A + I Simplex Method Applications The slack now in the system is the key. Each slack variable is associated with a constraint boundary Blais MIST 2017 Simplex Method MIST 2017 Blais Optimization Simplex algorithm: Linearity Higher- x Dimensional Split entries of into 2 subsets: Optimization y m Higher- Basic variables: xB 2 R (non-zero valued) Dimensional n Linearity Non-basic variables: xN 2 R (zero valued) Linear Represents a corner point of the feasible region. Programming Simplex Method If not optimal, move to an adjacent corner point by Applications swapping one entry in xB with one entry in xN and re-solving the system of equations. Blais MIST 2017 Simplex Geometry MIST 2017 Blais 1 0.9 Optimization 0.8 Linearity 0.7 0.6 Higher- 0.5 z Dimensional 0.4 Optimization 0.3 0.2 Higher- 0.1 Dimensional 0 1 Linearity 0.9 0.8 1 0.7 0.6 0.8 0.5 Linear 0.6 0.4 0.3 0.4 Programming 0.2 y 0.2 0.1 x Simplex Method 0 0 Applications Figure: Feasible Region Example: min f (x1; x2; x3) = ax1 + bx2 + cx3 subject to 5 linear constraints. Blais MIST 2017 Simplex Geometry MIST 2017 Blais 1 0.9 Optimization 0.8 Linearity 0.7 0.6 Higher- 0.5 z Dimensional 0.4 Optimization 0.3 0.2 Higher- 0.1 Dimensional 0 1 Linearity 0.9 0.8 1 0.7 0.6 0.8 0.5 Linear 0.6 0.4 0.3 0.4 Programming 0.2 y 0.2 0.1 x Simplex Method 0 0 Applications Example: min f (x1; x2; x3) = ax1 + bx2 + cx3 subject to 5 linear constraints.
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