Sela, Lina

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Utah State University DigitalCommons@USU All ECSTATIC Materials ECSTATIC Repository Spring 2020 Lecture Notes: Nonlinear Optimization and Matlab Optimization Toolbox Example Lina Sela The University of Texas at Austin, linasela@utexas.edu Follow this and additional works at: https://digitalcommons.usu.edu/ecstatic_all  Part of the Civil Engineering Commons Recommended Citation Sela, Lina, "Lecture Notes: Nonlinear Optimization and Matlab Optimization Toolbox Example" (2020). All ECSTATIC Materials. Paper 82. https://digitalcommons.usu.edu/ecstatic_all/82 This Lecture Material is brought to you for free and open access by the ECSTATIC Repository at DigitalCommons@USU. It has been accepted for inclusion in All ECSTATIC Materials by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. CE 385D Water Resources Planning and Management Spring 2020Lecture 09: Nonlinear optimization and Matlab optimization toolboxInstructor: Dr. Lina SelaModel summary:Here we will solve the example provided in Chapter 4.3.3, Daniel P. Loucks and Eelco van Beek2017, using Lagrange multipliers and Matlab optimization toolbox. The problem is to find theoperating and allocation policy that maximizes the net benefits. The optimization problem can beformulated as:Maximize (12− p1)p1 + (20− 1.5p2)p2 + (28− 2.5p3)p3−3p1.31 − 5p1.22 − 6p1.153subject to p1 ≤ 0.4x0.91p2 ≤ 0.5x0.82p3 ≤ 0.6x0.73x1 + x2 + x3 +R ≤ Qx1, x2, x3 ≥ 0p1, p2, p3 ≥ 0where xi the amount of water allocated to each firm i, pi is the amount of product produced byeach firm i, R is the amount that must remain in the river, and Q is the total amount of availablewater.Solution using Lagrange multipliers:Assuming that available water is a binding constraint and that all variables are positive, we canwrite the modified optimization problem:Maximize (12− p1)p1 + (20− 1.5p2)p2 + (28− 2.5p3)p3−3p1.31 − 5p1.22 − 6p1.153subject to p1 − 0.4x0.91 = 0 (λ1)p2 − 0.5x0.82 = 0 (λ2)p3 − 0.6x0.73 = 0 (λ3)x1 + x2 + x3 +R−Q = 0 (λ4)There are four equality constraints and we add a Lagrange multiplier λi for each constraint i. TheLagrangian function is:L(p, x, λ) = (12− p1)p1 + (20− 1.5p2)p2 + (28− 2.5p3)p3−3p1.31 − 5p1.22 − 6p1.153−λ1(p1 − 0.4x0.91)− λ2(p2 − 0.5x0.82)− λ3(p3 − 0.6x0.73)−λ4 (x1 + x2 + x3 +R−Q)}9-1Lecture 09: Nonlinear optimization and Matlab optimization toolbox 9-2Find the gradient of the Lagrange function and set it to zero, ∇L(p, x, λ) = 0.∂L∂p1= 0 12− 2p1 − 1.3× 3× p0.31 − λ1 = 0∂L∂p2= 0 20− 3p2 − 1.2× 5× p0.22 − λ2 = 0∂L∂p3= 0 28− 5p3 − 1.15× 6× p0.153 − λ3 = 0∂L∂x1= 0 λ1 × 0.9× 0.4× x−0.11 − λ4 = 0∂L∂x2= 0 → λ2 × 0.8× 0.5× x−0.22 − λ4 = 0∂L∂x3= 0 λ3 × 0.7× 0.6× x−0.33 − λ4 = 0∂L∂λ1= 0 p1 − 0.4× x0.91 = 0∂L∂λ2= 0 p2 − 0.5× x0.82 = 0∂L∂λ3= 0 p3 − 0.6× x0.73 = 0∂L∂λ4= 0 x1 + x2 + x3 +R−Q = 0These ten equations are the necessary conditions for an optimal solution. They can be solved toobtain the values of the ten unknown variables. We can solve them using the fsolve function.Steps to solve the equations:1. fsolve function solves a system of nonlinear equations of the form: F (x) = 0, where x is avector and F is a function that returns a vector value.2. Syntax: x = fsolve (fun,x0), where x0 is the initial guess and fun is a function handleor a function name that accepts a vector x and returns a vector F , the nonlinear equationsevaluated at x .3. Combine all variables into one vector x = [p1, p2, p3, x1, x2, x3, λ1, λ2, λ3, λ4].4. Write a function that computes the left-hand side of these ten equations in the form: F (x) = 0.Here for simplicity RQ stands for R-Q.Lecture 09: Nonlinear optimization and Matlab optimization toolbox 9-3function [F] = lagrange(x,RQ)F(1) = 12 - 2*x(1) - 1.3*3*x(1)ˆ0.3 - x(7);F(2) = 20 - 3*x(2) - 1.2*5*x(2)ˆ0.2 - x(8);F(3) = 28 - 5*x(3) - 1.15*6*x(3)ˆ0.15 - x(9);F(4) = x(7)*0.9*0.4*x(4)ˆ-0.1 - x(10);F(5) = x(8)*0.8*0.5*x(5)ˆ-0.2 - x(10);F(6) = x(9)*0.7*0.6*x(6)ˆ-0.3 - x(10);F(7) = x(1) - 0.4*x(4)ˆ0.9;F(8) = x(2) - 0.5*x(5)ˆ0.8;F(9) = x(3) - 0.6*x(6)ˆ0.7;F(10) = x(4) + x(5) + x(6) - RQ;end5. Define the initial guess:x0 = ones(10,1);6. Define RQ:RQ = 10;7. Define function handle:fun = @(x)lagrange(x,RQ); % note the the function has two inputs but only x ...is the vector of variables and RQ is a parameter8. Call the solver:x = fsolve(fun,x0);9. Compute the objective function: once we solved the equations we found the optimal solution.We can substitute the solution into the objective function and compute the optimal value.% compute objective functionf = (12 - x(1))*x(1) + (20 - 1.5*x(2))*x(2) + (28 - 2.5*x(3))*x(3) ...-(3*x(1)ˆ1.3 + 5*x(2)ˆ1.2 + 6*x(3)ˆ1.15);Lecture 09: Nonlinear optimization and Matlab optimization toolbox 9-410. The final code:clcclear% solving 10 nonlinear equations 4.35-4.44% x = [p1 p2 p3 x1 x2 x3 l1 l2 l3 l4]n = 10; % number of variablesRQ = 10; % available waterfun = @(x)lagrange(x,RQ);x0 = ones(n,1);x = fsolve(fun,x0);% compute objective functionf = (12 - x(1))*x(1) + (20 - 1.5*x(2))*x(2) + (28 - 2.5*x(3))*x(3) ...-(3*x(1)ˆ1.3 + 5*x(2)ˆ1.2 + 6*x(3)ˆ1.15);P = x(1:3);X = x(4:6);Lambda = x(7:10);function [F] = lagrange(x,RQ)F(1) = 12 - 2*x(1) - 1.3*3*x(1)ˆ0.3 - x(7);F(2) = 20 - 3*x(2) - 1.2*5*x(2)ˆ0.2 - x(8);F(3) = 28 - 5*x(3) - 1.15*6*x(3)ˆ0.15 - x(9);F(4) = x(7)*0.9*0.4*x(4)ˆ-0.1 - x(10);F(5) = x(8)*0.8*0.5*x(5)ˆ-0.2 - x(10);F(6) = x(9)*0.7*0.6*x(6)ˆ-0.3 - x(10);F(7) = x(1) - 0.4*x(4)ˆ0.9;F(8) = x(2) - 0.5*x(5)ˆ0.8;F(9) = x(3) - 0.6*x(6)ˆ0.7;F(10) = x(4) + x(5) + x(6) - RQ;endFinal solution: p1 = 0.46, p2 = 1.44, p3 = 1.88, x1 = 1.16, x2 = 3.74, x3 = 5.10, λ1 = 8.01, λ2 =9.24, λ3 = 11.03, λ4 = 2.84 and f = 53.56.Lecture 09: Nonlinear optimization and Matlab optimization toolbox 9-5Solution using Matlab optimization toolbox:We will solve the original optimization problem, i.e. with the inequality constraints. This is anonlinear optimization problem since the objective and constraint functions are nonlinear. Fornonlinear problems we’ll use the fmincon function.Maximize (12− p1)p1 + (20− 1.5p2)p2 + (28− 2.5p3)p3−3p1.31 − 5p1.22 − 6p1.153subject to p1 ≤ 0.4x0.91p2 ≤ 0.5x0.82p3 ≤ 0.6x0.73x1 + x2 + x3 +R ≤ Qx1, x2, x3 ≥ 0p1, p2, p3 ≥ 0Recall: The fmincon function finds the minimum of constrained nonlinear problem of the form:Min f(x)s.t. c(x) ≤ 0ceq = 0Ax ≤ bAeqx = beqlb ≤ x ≤ ubSyntax: [x,fval] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)Steps to transform equations:1. Separate bounds, nonlinear equalities, nonlinear inequalities, linear equalities, linear inequal-ities.2. Combine all variables into one vector x.3. Write vectors of lower and upper bounds: lb & ub such that lb ≤ x ≤ ub.4. Write matrix and vector of inequalities: A & b such that Ax ≤ b.5. Write matrix and vector of equalities: Aeq & beq such that Aeqx = beq.6. Write the function of the nonlinear constraints that accepts a vector x and returns two arrays,c(x) and ceq(x), where c(x) is the array of nonlinear inequality constraints at (x andceq(x) is the array of nonlinear equality constraints at x.7. Write the function of the nonlinear objective that needs to be minimized that accepts a vectorx and returns a real scalar f at x.8. Define initial guess x0.9. Call the solver: [x,fval] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)Lecture 09: Nonlinear optimization and Matlab optimization toolbox 9-6Complete the code:% Lagrange multiplier example 3.3 from the book% solve the optimization problem% Syntax: [x,fval,exitflag] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)clcclear% x = [p1 p2 p3 x1 x2 x3]n = 6; % number of variablesRQ = 10; % available water% lower bounds% linear inequality% nonlinear constraintsnonlcon = @mycon;% objective functionfun = @(x) -((12 - x(1))*x(1) + (20 - 1.5*x(2))*x(2) + (28 - 2.5*x(3))*x(3) ...-(3*x(1)ˆ1.3 + 5*x(2)ˆ1.2 + 6*x(3)ˆ1.15));x0 = ones(n,1);options = optimoptions('fmincon','Display','iter');[x,fval,exitflag, ,lambda] = fmincon( , , , , , , , , ,options)% constraint functionfunction [c,ceq] = mycon(x)c(1) =c(2) =c(3) =ceq =endFinal solution: p1 = 0.46, p2 = 1.44, p3 = 1.88, x1 = 1.16, x2 = 3.74, x3 = 5.10, the Lagrangemultipliers are: λ1 = 8.00, λ2 = 9.24, λ3 = 11.03, λ4 = 2.84, and the optimal value is f = 53.47.

Lecture Notes: Nonlinear Optimization and Matlab Optimization Toolbox Example

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Lecture Notes: Nonlinear Optimization and Matlab Optimization Toolbox Example

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