Figure : An 8 bridge design grid. (a) Run this model using LOQO. What is the otimal comliance? What is the running time?

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1 5.094/SMA53 Systems Otimization: Models and Comutation Assignment 5 (00 o i n ts) Due Aril 7, 004 Some Convex Analysis (0 o i n ts) (a) Given ositive scalars L and E, consider the following set in three-dimensional sace: ( ) L S = (f; t; s) j f» s t ; s 0 ; t 0 : E This set arises in the otimization of load bearing truss structures that you have seen in the lectures on truss design. Show that S is a convex set. (b) Let S k k denote the set of k k square symmetric matrices. A matrix X S k k is symmetric ositive semidefinite (SPSD) if v T k Xv 0 for all v < : Let denote the set of k k SDSD matrices. Prove that S k k S k k + + is a convex set. (c) A matrix X S k k is symmetric ositive definite (SPD) if v T X v > 0 for all v < k ; v 6= 0 : Let denote the set of k k SPD matrices. Prove that S k k S k k is a convex set. Truss Design (0 o i n ts) The urose of this roblem is to give y ou some exerience in solving truss design models using AMPL and LOQO. Figure shows an 8 grid for a bridge design roblem. The left and right bottom oints in the grid, which are the ordered airs (0; 0) and (0; 0), are fixed oints in the grid. There is a downward force at each grid oint on the bottom row of the grid. The AMPL model file bridge8soc.mod contains a truss design model of this bridge design roblem (modeled using the second-order cone formulation discussed in class) that uses the data file basic8.dat.

2 Figure : An 8 bridge design grid. (a) Run this model using LOQO. What is the otimal comliance? What is the running time? How m a n y iterations did the algorithm take? (b) Create and hand in a icture of the otimal solution using the MATLAB file resm.m. (c) In reality, structures need to b e designed to handle many different external forces. For examle, the bridge should b e designed to handle forces from winds in various directions that may shift over time as well as vary in intensity. In order to see how sensitive the otimal solution might b e to different external forces, we ask you to try several different external force (load) rofiles on the bottom of the bridge. As a common basis of comarison use the following external loads: (i) F = ; ; ; ; ; ; ; ; (ii) F = ; ; ; ; ; ; ; ; (iii) F = ; ; ; ; ; ; ; ; (iv) F = ; ; ; ; ; ; ; ; ψψ!! (v) F = ; ; ; ; ; ; ; ; (c) What do you observe? How might y ou take l o a d v ariations into account when solving this roblem in ractice?

3 3 Truss Design Formulations (0 o i n ts) In this roblem, you are asked to exlore ossible changes in the the model arameters and in the formulation for the bridge design model shown in Figure. (a) Maximum Volume Constraint. The current total volume constraint uer bound is,600.0 units. Change this bound to values between 00.0 units and 00,000.0 units and solve the resulting model. Record the otimized comliance and the numb e r of iterations it takes to solve the model as functions of the volume constraint uer bound. What do you observe? Can you suggest any reasons for this? (b) Tolerance Parameter ffl. If the value of the Young's modulus is E = :0, then the second-order cone constraint in the truss design roblem is of the form: q L f + y» w : However, you will notice in the AMPL file bridge6soc.mod that the constraint i n t h e model is actually: q ffl + L f + y» w ; where ffl is a tolerance arameter given in the inut data file basic6.dat. The value of ffl given in basic6.dat is ffl = 0 : If ffl = 0 :0, then the AMPL model is the same as the true formulation of the roblem. If ffl > 0, then the AMPL model will have a slightly smaller feasible region than the true model of the roblem. The urose of introducing ffl > 0 is to seed u solution time by smoothing" the second-order cone constraint a b i t. Change the value of the tolerance arameter ffl in the range from ffl = 0 : u to ffl = :0 and solve the resulting model. Record the otimal comliance and the number of iterations it takes to solve the model as functions of the value of ffl. What do you observe? Can you suggest any reasons for this? 4 High-Percentage Covering Disk Problem (0 o i n ts) Consider the roblem of determining the smallest disk that contains the m given oints c ; : : : ; c m. The decision variables in the roblem are the center x and the radius R of the enclosing disk, and the roblem yields the simle formulation: 3

4 CDP : minimize x;r R s.t. kx c i k» R i = ; : : : ; m n x < ; R < ; where in this case n = is the dimension in which the roblem arises. The Aml file CDPql.mod contains a reformulation of this roblem with quadratic objective and linear constraints. CDPql.mod is the same formulation but demonstrates the roblem command in Aml. The Aml file CDPlq.mod contains the version of the roblem with a linear objective and quadratic constraints. All of these formulations of the roblem solve v ery raidly, with zero duality g a. (a) The data files cover8.dat and cover9.dat each contain the locations of 00 oints in n = dimensions. The first column in the file is the index of the o i n t, and the second and third columns contain the first and second coordinates of the oint. Solve the roblem using these data sets, and create figures illustrating your solution. Solve CDPql.mod with the LOQO convex otion turned on and off. Is there a difference in how LOQO solves the roblem? If so, why? Solve CDPlq.mod with the LOQO convex otion turned on and off. Is there a difference in how LOQO solves the roblem? If so, why? (b) Consider the following variation of the roblem, where we would like to find the smallest disk that contains 90% of the oints. This roblem arises in data mining, for examle. We will use the sigmoid function: f ff (s) := + e ffs to account for o i n ts that might lie outside the disk. The sigmoid function with arameter ff > 0 has the following attractive roerties: f ff (s)! 0 a s s! f ff (s)! a s s! + f ff (s) = f ff ( s) f ff (0) = A grah of this function is shown in Figure. Using the sigmoid function, our roblem can be aroximated as the following smooth nonlinear roblem: HPCDP : minimize x;r;s ;:::;sm R s.t. kx c i k» R + s i i = ; : : : ; m n x < ; R < ; s ; : : : ; s m < P m f ff (s i )» 0:m : i= 4

5 Figure : The sigmoid function. In class we considered a reformulation of this roblem where we squared the constraint kx c i k» R+s i " after adding the relaxation variables s i, yielding the new constraint (x c i ) T (x c i )» R + Rs i + s i ". Instead of this aroach, create a reformulation where you square the constraint kx c i k» R" and then add the relaxation variables s i, namely (x c i ) T (x c i )» R + s i ". Modify the AMPL model from art (a) and solve HPCDP using the data files cover8.dat and cover9.dat. Comute solution results for ff = 7, 0, and 0. What do you observe? 5 Disk and Sherical Packing Problem (0 o i n ts) Consider the roblem of acking a variety of wires into a cable. The m wires will have given radii r ; r ; : : : ; r m. We w ould like to determine the minimum width of a cable that will be used to enclose the wires. We can concetualize this roblem by considering a cross-section of the cable. The decision variables in the roblem are the centers x ; : : : ; x m of the m disks of radii r ; : : : ; r m, and the radius R of the disk that is the cross-section of the enclosing cable. This roblem has the following formulation: PP : minimize x ;:::;xm;r R s.t. kx i x j k r i + r j i = ; : : : ; m ; j = i + ; : : : ; m kx i k + r i» R i = ; : : : ; m n x ; : : : ; x m < ; R < ; where in this case n = is the dimension in which the roblem arises. The first set of constraints ensures that no two wires overla the same sace, and the second set of constraints ensures that the cable encloses all of the wires. (a) One roblem with the formulation of PP is that the constraint functions are not differentiable. Convert this formulation into an equivalent form where all functions 5

6 6 are differentiable. (b) Construct an iterative AMPL model to solve the roblem. Do not use the model PP.com that we designed and resented in class; instead we ask you to develo your own iterative AMPL model. The data files ack8.dat and ack8a.dat contain radius data for m = 60 radii of wires. In each of these files, the first column contains the index (;, etc.), the second column contains the number of oints of that index, and the third column contains the radius associated with that index. For examle, suose that the file data is as shown in Table. This indicates that there are 50 wires of radius 0.0, also 40 wires of radius 5.0, and also 0 wires of radius 8.7 to be acked. Index Number of Wires with given Radius Radius Table : Examle of a acking data file. Describe your iterative solution method. Use your model to solve the roblem for these two data sets, and create figures illustrating your final comuted solution.