# American Institute of Mathematical Sciences

December  2017, 10(6): 1329-1350. doi: 10.3934/dcdss.2017071

## Inverse truss design as a conic mathematical program with equilibrium constraints

 1 School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom 2 Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic

* Corresponding author

Dedicated to Tomáš Roubíček on the occasion of his 60th birthday

Received  July 2016 Revised  December 2016 Published  June 2017

Fund Project: The second author is supported by the Grant Agency of the Czech Republic project 15-00735S

We formulate an inverse optimal design problem as a Mathematical Programming problem with Equilibrium Constraints (MPEC). The equilibrium constraints are in the form of a second-order conic optimization problem. Using the so-called Implicit Programming technique, we reformulate the bilevel optimization problem as a single-level nonsmooth nonconvex problem. The major part of the article is devoted to the computation of a subgradient of the resulting composite objective function. The article is concluded by numerical examples demonstrating, for the first time, that the Implicit Programming technique can be efficiently used in the numerical solution of MPECs with conic constraints on the lower level.

Citation: Michal Kočvara, Jiří V. Outrata. Inverse truss design as a conic mathematical program with equilibrium constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1329-1350. doi: 10.3934/dcdss.2017071
##### References:

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##### References:
Two-bar truss
Two solutions of the $3\times 3$ truss design problem with all nodes connected
Five-by-three truss (Ex. 2): initial layout and optimal design
Five-by-five truss (Ex. 3): initial layout and optimal design
Five-by-five truss (Ex. 3): optimal load and corresponding design as computed by BTNCLC with INI1 (left) and INI3 (right)
Results of Example 2. The last columns show the number of iterations needed to obtain the value of the objective function smaller than $10^{-3}$-$10^{-6}$ and (the last column) to fulfill the stopping criterion
 method INI $c^*$ number of iterations to reach $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ $c^*$ BTNCLC 1 $2.4\cdot 10^{-8}$ 23 36 50 63 88 BTNCLC 2 $2.0\cdot 10^{-8}$ 6 15 25 43 64 BTNCLC 3 $7.6\cdot 10^{-8}$ 9 11 16 23 26 NMSMAX 1 $7.7\cdot 10^{-6}$ 547 1018 3631 - 6617 NMSMAX 2 $1.5\cdot 10^{-4}$ 694 - - - 4406 NMSMAX 3 $5.7\cdot 10^{-6}$ 33 66 106 - 192
 method INI $c^*$ number of iterations to reach $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ $c^*$ BTNCLC 1 $2.4\cdot 10^{-8}$ 23 36 50 63 88 BTNCLC 2 $2.0\cdot 10^{-8}$ 6 15 25 43 64 BTNCLC 3 $7.6\cdot 10^{-8}$ 9 11 16 23 26 NMSMAX 1 $7.7\cdot 10^{-6}$ 547 1018 3631 - 6617 NMSMAX 2 $1.5\cdot 10^{-4}$ 694 - - - 4406 NMSMAX 3 $5.7\cdot 10^{-6}$ 33 66 106 - 192
Results of Example 3. The last columns show the number of iterations needed to obtain the value of the objective function smaller than $10^{-3}$-$10^{-6}$ and (the last column) to fulfill the stopping criterion
 method INI $c^*$ number of iterations to reach $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ $c^*$ BTNCLC 1 $2.6\cdot 10^{-7}$ 67 92 111 313 143 BTNCLC 2 $4.8\cdot 10^{-8}$ 9 29 46 71 99 BTNCLC 3 $4.4\cdot 10^{-1}$ - - - - 11 NMSMAX 1 $5.3\cdot 10^{-3}$ - - - - 13487 NMSMAX 2 $1.7\cdot 10^{-2}$ - - - - 7850 NMSMAX 3 $2.0\cdot 10^{-6}$ 68 157 171 - 207
 method INI $c^*$ number of iterations to reach $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ $c^*$ BTNCLC 1 $2.6\cdot 10^{-7}$ 67 92 111 313 143 BTNCLC 2 $4.8\cdot 10^{-8}$ 9 29 46 71 99 BTNCLC 3 $4.4\cdot 10^{-1}$ - - - - 11 NMSMAX 1 $5.3\cdot 10^{-3}$ - - - - 13487 NMSMAX 2 $1.7\cdot 10^{-2}$ - - - - 7850 NMSMAX 3 $2.0\cdot 10^{-6}$ 68 157 171 - 207
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