How to solve optimization problems when the constraints have derivative (symbolic) functions

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Jen W 2020-12-7

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评论： Jen W 2020-12-8

Hello,

I am trying to solve an optimization problem as following. There are two parabolas f and g. When they have a common tangent line, I want the common tangent points on these two curves to be as close to x1 and x2 in the parabolic functions as possible (see f and g functions in the code below). I have defined optimvar, constraints, and objective in the code. I know the main issue of the code is that I am mixing symbolic variable (x) and optimvar in functions f and g, but I do not know how to approach this in another way. I have tried to use function handles and matlabFunction, but I cannot seem to decouple symbolic variables and optimvar. Could someone please help me? Any idea will be appreciated. Thank you.

syms x 
x1_tan = 3; % x-coordinate of the tangent point on f
x2_tan = 6; % x-coordinate of the tangent point on g
prob = optimproblem('ObjectiveSense', 'minimize');
a = optimvar('a', 'LowerBound', 0, 'UpperBound', 100);
b = optimvar('b', 'LowerBound', 0, 'UpperBound', 100);
x1 = optimvar('x1', 'LowerBound', 0, 'UpperBound', 10);
x2 = optimvar('x2', 'LowerBound', 0, 'UpperBound', 10);
f = a*(x - x1)^2 + 10; % this won't work because it mixes syms and optimvar
g = b*(x - x2)^2 + 2; % this also won't work, same reason
% constraints to make sure that the tangent points x-coordinates are x1_tan and x2_tan
prob.Constraints.eq1 = subs(diff(f, x), x, x1_tan) == subs(diff(g, x), x, x2_tan); % tangent line slopes are the same
prob.Constraints.eq2 = subs(diff(f, x), x, x1_tan)*x1_tan - subs(f, x, x1) == subs(diff(g, x), x, x2_tan)*x2_tan - subs(g, x, x2); % tangent line y-axis intercepts are the same
prob.Objective = sqrt((x_tan - x1)^2 + (y_tan - x2)^2); % the goal is to minimize this function

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Answer 1

Walter Roberson 2020-12-7

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f = a*(x - x1)^2 + 10; % this won't work because it mixes syms and optimvar
g = b*(x - x2)^2 + 2; % this also won't work, same reason

Use symbolic X X1 X2 for f and g. Take the derivative. subs any constant. matlabFunction with vars [X X1 X2] to get a numeric function. Now pass in optimvar variables to get out an optimization expression to use. That is, I expect that you will need an additional optimization variable named x

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Walter Roberson 2020-12-8

You have to pass in the options

https://www.mathworks.com/help/optim/ug/optim.problemdef.optimizationproblem.solve.html#namevaluepairarguments

Jen W 2020-12-8

Worked. Appreciate your help.

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Answer 2

Matt J 2020-12-7

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编辑：Matt J 2020-12-7

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The calculus needed to take the derivatives in your constraint expressions seems pretty trivial. Do you really need to go through the Symbolic Toolbox at all?

prob.Constraints.eq1 = 2*a*(x1_tan-x1) == 2*b*(x2_tan-x2) 
prob.Constraints.eq2 = 2*a*(x1_tan-x1)*x1_tan - 10 == 2*b*(x2_tan-x2)*x2_tan - 2; 
prob.Objective = (x_tan - x1)^2 + (y_tan - x2)^2; % omit the square root