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of the Global Optimization Toolbox\251 Maplesoft, a division of Waterloo Maple Inc., 2004The intent of this application example is to illustrate Maple techniques in a real world application context. Maple is a general-purpose environment capable of solving problems in any field that depends on mathematics and data. This application illustrates one possibility for this particular field. Note that there are many options within the Maple system to optimize the computations for specific problems.

The algorithms in the Optimization package that comes with base Maple 9.5 are local search methods. Local search methods use first and second derivatives to find a local extremum, iteratively producing points that improve the objective function while maintaining or approaching feasibility. The algorithms in the Global Optimization Toolbox are global search methods. Instead of using derivatives to find a local extremum, they systematically search the entire feasible region for a global extremum. Global search methods find a global extremum in optimization models that have many local extrema, and they do not rely on an initial point.This application illustrates the following features and benefits of Maple: \342\200\242 The power of the Global Optimization Toolbox \342\200\242 The ease and flexibility of linear algebra computations \342\200\242 Use of color contours to highlight the local extrema of 3-D plots \342\200\242 Gaining insight into engineering problems symbolically, numerically and visually \342\200\242 The symbolic differential equation solvers

restart: interface( warnlevel=0 ): interface(displayprecision=4): with( plots ): with( Optimization ): with( GlobalOptimization ): with( LinearAlgebra ):

Find the global minimum of this quartic polynomial over the interval [0, 5]. Note that the function is nonconvex and has two local mimima.poly := (x-1)*(x-2)*(x-3.5)*(x-4.8);plot( poly, x=0..4.8, view=[0..5,-4..8] );Depending on the starting point provided, the built-in local NLP solver returns the global minimum at x=4.298 or the sub-optimal local minimum at x=1.407. NLPSolve( poly, initialpoint=[x=5] ); NLPSolve( poly, initialpoint=[x=3] ); NLPSolve( poly, initialpoint=[x=2] ); NLPSolve( poly, initialpoint=[x=0] );The GlobalOptimization solver always find the global minimum. Note that the calling sequence requires no starting point but does require a finite range for x, which is set to 0..5.GlobalSolve( poly, x=0..5 );

Find the global minimum of a 2-D nonconvex quadratic function over a rectangle. Construct the function from a non-positive-definite matrix. Q := Matrix( <<1.9,-3.5>|<5,-1/3>> );Check that the matrix is not positive definite.IsDefinite( Q );Construct the function NiMqKCklInhHJSJ0RyIiIiUiUUdGJ0YlRic=NonconvexQuadratic := expand( Transpose(<x,y>) . Q . <x,y> );Plot the function and its contours. There are global minima at [3.947, -10.000] and [-3.947, 10.000], and a saddle point at [0, 0].p1 := plot3d( NonconvexQuadratic, x=-10..10, y=-10..10, axes=boxed, transparency=.7, style=patch): p2 := plot3d( -100, x=-10..10, y=-10..10, style=patchnogrid, color=NonconvexQuadratic ): plots[display]( p1, p2, orientation=[125,60] ); From most starting points, the Maple local solver finds the global minimum at [3.947, -10.000].NLPSolve( NonconvexQuadratic, x=-10..10, y=-10..10, initialpoint=[x=.1, y=0]);If the starting point is chosen poorly, the local solver stops at the saddle point [0, 0]. This is because the local solver terminates if it detects a zero gradient at its current point. Saddle points have zero gradients but are not extrema.NLPSolve( NonconvexQuadratic, x=-10..10, y=-10..10, initialpoint=[x=0, y=0]);The global solver always finds one of the two global minima. Again, finite ranges for x and y are provided. GlobalSolve( NonconvexQuadratic, x=-10..10, y=-10..10);

Find the global minimum of a 2-D convex function over a plane minus the unit disk, which is a nonconvex region.obj := x^2 + y^2 - x + y;constraint := x^2 + y^2 >= 1;The global minimum is at [.707, -.707], shown in the following plot in red at the boundary of the feasible region. Several contours of the objective function are shown. Starting from the point [-2, 2], the Maple local solver finds the non-optimal local minimum at [-.707, .707], shown in blue. The gradient of the objective function at [-.707, .707] is perpendicular to the direction the search would need to move next to get around the infeasible region, terminating the local search.p1 := plottools[disk]([0,0],1,color=green): p2 := plots[contourplot](obj, x=-2..2, y=-2..2, coloring=[blue,red], contours=20): p3 := plots[pointplot]([[.707,-.707]],color=red, symbolsize=20, symbol=circle): p4 := plots[pointplot]([[-.707,.707],[-2,2] ],color=blue, symbolsize=20, symbol=circle): plots[display]( p1, p2, p3, p4 ); Local search from [-2, 2] finds a suboptimal local minimumNLPSolve( obj, {constraint}, x=-2..2, y=-2..2, initialpoint=[x=-2,y=2]);Global search finds the global minimum at [.707, -.707].GlobalSolve( obj, {constraint}, x=-2..2, y=-2..2);

Consider a 4-D function from optimization benchmark tests. This function has an extremely flat, but non-constant, region. Local search methods find the flat region, but terminate before finding the global minimum because the gradient of the function in that region is nearly zero.objWood := 100*(y-x^2)^2+(1-x)^2+90*(z-w^2)^2+(1-w)^2+10.1*((y-1)^2+(z-1)^2)+19.8*(y-1)*(z-1);The global minimum is at x = y = z = w = 1. To see why this is a difficult function to minimize, consider its 2-D projection fixing z=1 and w=1.objWoodXY := eval( objWood, { z=1, w=1 } );The contours indicate that the function is extremely flat around the origin. Local search methods generally find a local minimum in this region and terminate.p1 := plot3d( objWoodXY+800, x=-2..2, y=-2..2, axes=boxed, transparency=.5, style=patch): p2 := plot3d(0, x=-2..2, y=-2..2, style=patchnogrid, color=objWoodXY): plots[display](p1,p2, orientation=[39,72] ); The Maple local search terminates before finding the global minimum, but the global search finds the minimum easily.NLPSolve( objWood, initialpoint=[x=-2,y=2,z=-2,w=3] ); GlobalSolve( objWood, x=-10..10, y=-10..10, z=-10..10, w=-10..10 );

The examples in the previous section are simple exercises that demonstrate how global search methods can greatly outperform local search methods. Following are some real-world applications of global optimization.

Consider the problem of designing a suspension system that exhibits some specified behavior in response to a bump in the road. The problem variables are the spring constant k and the damper constant b. Given the mass of the car on each wheel, m, and the expected amplitude of a typical bump, find values for k and b to create a system response that matches the desired response. The objective function to be minimized is the squared error between the desired and actual response as a function of k and b over a discrete set of times. After deriving the actual response by solving the system's differential equation, use optimization to find the values of k and b that minimize the error.

Suppose the target response is a decaying exponential function. When the automobile hits a bump of amplitude 0.1 meter, the amplitude of its oscillations decays exponentially.restart:Target := t->(0.14*exp(-4.9*t)*cos(5*t -.7754)):plot( Target(t), t = 0..1, title="Target Response", labels=["seconds","metres"], thickness=2 );

To derive the actual response of the system to a bump, solve the differential equation of the system's behaviour with the initial condition x(0) = 0.1.The differential equation of an unforced mass-spring-damper system:System_Equation := m*diff(x(t),t,t) + b*diff(x(t),t) + k*x(t) = 0;Now solve the system and define the response as a function of k, b and t, fixing m=450 kg.m:=450:sol := dsolve( {System_Equation, x(0)=.1, D(x)(0)=0} );The actual response of the system as a function of k, b and t:Actual := unapply( rhs( sol ), k, b, t):Overlay plots of the actual response with k=10^4 N/m and b=10^3 Ns/m, and the target response. plot([ Target(t), Actual(10^4, 10^3, t) ], t=0..1, thickness=3, labels=["seconds","metres"], title="Target and Actual responses for k = 10^4 and b = 10^3" ); These choices of k and b do not provide a good match to the target. One method for improving the match is to substitute different combinations of k and b until an acceptable solution is found. A more systematic approach using optimization is described in the next section.

Ideally, error is measured as the integral of the squared difference between the desired and actual response. However, evaluating the integral in this application is extremely difficult, due to the complexity of the response function. As an approximation, sample the functions at key times and add the squared differences between the functions at these points.Sample the output at times NiYvJSJ0RyomIiIiRiYiIiMhIiJGJiomIiIkRiZGJ0YoRic=time_sample:= 1/2, 1, 3/2, 2;The objective function is the sum of squared differences between the Actual and Target responses at these four times. Although the error function has only two variables, it is extremely complicated.Error := add( (Actual(k, b, time_sample[i]) - Target(time_sample[i]))^2, i = 1..4);Plot the error function over the k-b plane. The function is not only non-convex, but also flat in the region that the global minimum is likely to be found. It is not known before computation whether the plot region shown contains the desired solution. (Note: according to the theory of differential equations, the imaginary terms of the response function, and therefore the error function, always cancel. However, the Maple plot function does not verify this before calculation. To plot the function correctly, plot the real part of the error function, which is identical to the function itself.) plot3d( Re(Error), b=100..5000, k=1000..25000, axes=boxed, shading=zhue, transparency=.35, style=patchnogrid, orientation=[75,75] );

Minimize the error functionwith( GlobalOptimization );The toolbox finds the global minimum. Notice that the error is very close to zero, as desired.sol := GlobalSolve( Re(Error), b=100..10000, k=100..40000 );Now plot the error function in a neighborhood of the global minimum. Even in this small region, it is not clear from the plot alone where the global minimum lies.plot3d( Re(Error), b=4300..4600, k=21000..23000, axes=boxed, shading=zhue, transparency=.35, style=patchnogrid, orientation=[75,75] );

Maple found the global minimum at k = 22.691 kN/m, b = 4.531 kNs/m. Now overlay the desired and actual plots. They match perfectly. Plotting the difference between target and actual, the greatest difference is NiMqJiIiIyIiIikiIzUsJCIiJiEiIkYl m, only 20 micrometres.assign(sol[2]);plot([Target(t), Actual(k, b, t)+.05, .05 ], t=0..1, labels=["seconds","metres"], title = "Target and System Responses Overlaid (with vertical offset)", thickness=2, color=[red,green,black]);The absolute value of the difference between the target and actual responses:plot(abs(Target(t) - Actual(k, b, t)), t=0..1, title="Absolute Value of Target minus System Response", labels=["seconds","meters"], thickness=2);

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. 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