Alternating direction method of multipliers for truss topology optimization with limited number of nodes: a cardinality-constrained second-order cone programming approach

Abstract

This paper addresses the compliance minimization of a truss, where the number of available nodes is limited. It is shown that this optimization problem can be recast as a second-order cone programming with a cardinality constraint. We propose a simple heuristic based on the alternative direction method of multipliers. The efficiency of the proposed method is compared with a global optimization approach based on mixed-integer second-order cone programming. Numerical experiments demonstrate that the proposed method often finds a solution having a good objective value with small computational cost.

Layout optimization of simplified trusses using mixed integer linear programming with runtime generation of constraints

“…The modelling of (3a) is similar to that found in the literature (e.g. Kanno and Fujita 2018) when only the number of nodes are limited. However in light of the presence of integer flags for member existence in the present formulation, an equivalent formulation whereby v and w are linked is possible.…”

“…Additionally, significant attention has been devoted to limiting the numbers of different cross sections present in a given solution. Kanno and Fujita (2018) limit the number of joints in solutions whilst minimizing the compliance of the structure, considering both a heuristic method and a mathematical programming formulation including integer variables. However, although the resulting mixed integer second-order cone programming (MISOCP) problem could solve problems with up to 1500 potential members reasonably quickly (in a time of 112 s), conditions were not imposed to prevent intersecting or overlapping members.…”

Exploiting Lagrange duality for topology optimizationwith frictionless unilateral contact

“…Unlike the convex optimization approach in Kočvara et al [30] for trusses, the SOCP formulation presented in this paper retains the member cross-sectional areas as optimization variables. Therefore, various additional design constraints, e.g., limitation of the number of different cross-sectional areas [17] and limitation of the number of nodes [20], as well as the self-weight load [23], can be incorporated into this SOCP formulation. Problems with such constraints are formulated as MISOCP (mixed-integer second-order cone programming) problems, which can be solved globally with a standard mixedinteger programming solver.…”

“…where the vertices of X have been considered to evaluate the optimal value of a linear programming problem. Substitution of (21) into (20) yields the following optimization problem:…”

“…Associated with each truss member, we introduce a binary variable to indicate whether the member exists or vanishes. Then, we can consider various design constraints, including lower bound constraints on the cross-sectional areas of existing members [23], an upper bound constraint on the number of nodes [20], limitation of the number of different values of cross-sectional areas [17,19], and upper bound constraints on the degrees of nodes [21]. All of these constraints can be treated within the framework of mixed-integer second-order cone programming (MIS-OCP).…”