CS/ECE/ISyE 524 Homework 8: Convex Optimization

1. Heat pipe design. A heated uid at temperature T (degrees above ambient temperature) ows in a pipe with xed length and circular cross section with radius r. A layer of insulation, with thickness w, surrounds the pipe to reduce heat loss through the pipe walls (w is much smaller than r). The design variables in this problem are T , r, and w.

The energy cost due to heat loss is roughly equal to ₁T r=w. The cost of the pipe, which has a xed wall thickness, is approximately proportional to the total material, i.e., it is given by ₂r. The cost of the insulation is also approximately proportional to the total insulation material, i.e., roughly ₃rw. The total cost is the sum of these three costs.

The heat ow down the pipe is entirely due to the ow of the uid, which has a xed velocity, i.e., it is given by ₄T r². The constants _i are all positive, as are the variables T , r, and w.

Now the problem: maximize the total heat ow down the pipe, subject to an upper limit C_max on total cost, and the constraints

T_min T T_max; r_min r r_max w_min w w_max; w 0:1r

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1. Express this problem as a geometric program, and convert it into a convex optimization problem. Recall that a generic geometric program has the following form

Consider a simple instance of this problem, where C_max = 500 and ₁ = ₂ = ₃ = ₄ = 1. Also assume for simplicity that each variable has a lower bound of zero and no upper bound. Solve this problem using JuMP. Use the Ipopt solver and the command @NLconstraint(...) to specify nonlinear constraints such as log-sum-exp functions. What is the optimal T , r, and w?