Inverse truss design as a conic mathematical program with equilibrium constraints

Michal Kočvara; Jiří V. Outrata

doi:10.3934/dcdss.2017071

Article Contents

2017, Volume 10, Issue 6: 1329-1350. Doi: 10.3934/dcdss.2017071

This issue Previous Article Averaging of time-periodic dissipation potentials in rate-independent processes Next Article Two boundary value problems involving an inhomogeneous viscoelastic solid

Inverse truss design as a conic mathematical program with equilibrium constraints

Michal Kočvara^1,2, , and
Jiří V. Outrata^2,

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
* Corresponding author

Received: June 30, 2016

Revised: November 30, 2016

Published: December 2017

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

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Abstract

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.

Keywords:

Mathematical programs with equilibrium constraints,
conic optimization,
truss topology optimization.

Mathematics Subject Classification: Primary: 49K40, 74P05; Secondary: 49M05, 90C30.

Citation:

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Figure 1. Two-bar truss

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Figure 2. Two solutions of the $3\times 3$ truss design problem with all nodes connected

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Figure 3. Five-by-three truss (Ex. 2): initial layout and optimal design

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Figure 4. Five-by-five truss (Ex. 3): initial layout and optimal design

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Figure 5. Five-by-five truss (Ex. 3): optimal load and corresponding design as computed by BTNCLC with INI1 (left) and INI3 (right)

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Table 1. 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
method	INI	$c^*$	$10^{-3}$	$10^{-4}$	$10^{-5}$	$10^{-6}$	$c^*$
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

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Table 2. 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
method	INI	$c^*$	$10^{-3}$	$10^{-4}$	$10^{-5}$	$10^{-6}$	$c^*$
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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