Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 61: (DOI: /nme.1112) Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound M. Stolpe, Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, DK-2800 Kgs. Lyngby, Denmark SUMMARY We present a convergent continuous branch-and-bound algorithm for global optimization of minimum weight truss topology problems with displacement, stress, and local buckling constraints. Valid inequalities which strengthen the problem formulation are derived. The inequalities are generated by solving well-defined convex optimization problems. Computational results are reported on a large collection of problems taken from the literature. Most of these problems are, for the first time, solved with a proof of global optimality. Copyright 2004 John Wiley & Sons, Ltd. KEY WORDS: topology optimization; stress constraints; global optimization 1. INTRODUCTION We consider global optimization of truss topology optimization problems with displacement, stress, and local Euler buckling constraints. The objective function is the weight of the structure and the variables are the cross-section areas of the bars, the nodal displacements, the member stresses, and the member forces. The non-linear constraints consist of bilinear equilibrium equations and quadratic buckling (stability) constraints. The stress and displacement constraints are modelled as linear inequality constraints. The feasible set of these problems is, in general, non-convex and normally there exist several local minima which are not global. We propose to solve these problem using a continuous branch-and-bound method which guarantees convergence to a global optimal solution. Several methods based on branch-and-bound have been developed for general continuous optimization problems involving non-convexities in the objective function and constraints. Among these are αbb, see e.g. References [1, 2], and the branch-and-reduce algorithm BARON, see Reference [3]. There are also specialized algorithms for non-convex problems with quadratic Corresponding author. M. Stolpe, Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, DK-2800 Kgs. Lyngby, Denmark. M.Stolpe@mat.dtu.dk Received 2 December 2002 Revised 8 January 2004 Copyright 2004 John Wiley & Sons, Ltd. Accepted 10 March 2004

2 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1271 objective functions and quadratic constraints such as the branch-and-cut method presented in Reference [4] and the decomposition method GOP, see References [5, 6]. A branch and contract algorithm for problems with bilinear and linear fractional terms is presented in Reference [7]. For an introduction to branch-and-bound and other deterministic methods in global optimization, see e.g. References [8, 9]. In Reference [10], a branch-and-bound method for truss topology optimization problems with stress and displacement constraints was presented. In that method a set of candidate trusses are examined. To obtain a lower bound on the weight for a given topology, all stress constraints are removed and the resulting non-linear program is solved. The method is not guaranteed to find the global optimum of the problem since the lower bounding problems are non-convex. Simões [11], also proposes to solve truss topology optimization problems with stress and displacement constraints using a continuous branch-and-bound method. By exploring the bilinearity in the constraints, convex lower bounding problems are constructed and solved at every node in the search tree. However, very limited computational results are reported. There is an important generalization of the local buckling constraints which is not considered in this paper. The local buckling constraints should take into account the effects of the so-called chains which often appear in the optimal solution of truss topology design problems. A chain consists of a straight connection of bars where the inner nodes are not connected to any other bars in the structure. If a chain with members in compression appears then the local buckling constraints should be evaluated with the length of a single bar which replaces the chain. These so-called topological stability constraints are modelled in Reference [12]. Another important topic in truss topology design, which is not considered in this paper, is the issue of global stability of the structure. If the stability is modelled using the so-called linear buckling model, then a constraint on the global stability of the structure may be modelled as a non-convex semidefinite matrix inequality, see References [13, 14]. If the structure is subject to only one load condition and only stress and local buckling constraints are included, the considered problem enjoys special structure. In this case, it has been shown that there exists an optimal solution which corresponds to a statically determinate structure and therefore the compatibility equations and the displacement variables may be removed from the formulation. For this situation special methods have been developed, see e.g. Reference [15]. In Reference [12] the single load model is extended to include topological stability constraints and slenderness constraints. A method for this extended problem based on solving a sequence of linear programs is presented in Reference [16]. However, none of the above methods developed for the single load case will in general converge to a global optimal solution since they only take local information into account. If the structure is subject to only one load condition and only stress constraints are considered the situation becomes very favourable. In this case the problem may equivalently be written as a linear program, see e.g. References [17, 18]. The considered truss topology optimization problem may equivalently be modelled using only the cross-section areas as variables, the so-called nested formulation. In this case, it has been shown that the feasible set may be disconnected. This can for instance happen if local buckling constraints are considered or if upper bounds on the area variables are present, see e.g. References [19, 20]. Furthermore, the feasible set may contain degenerate parts with zero measure. Typically the global optimal solution is located in one of these degenerate parts, so called singular topologies, see e.g. References [19, 21 24]. Therefore, basic constraint qualifications generally do not hold at the global optimum. It is indicated in e.g. Reference

3 1272 M. STOLPE [22] that this phenomenon is caused by the discontinuity of the stress constraints at zero area. The stress may approach a finite non-zero value when the area of a bar approaches zero, while the stress constraint itself should be removed when the area is zero. To overcome this difficulty, it has been suggested that the stress constraints should be approximated or perturbed so that the degenerate parts are expanded. Rozvany and Sobieszczanski-Sobieski [25] and Rozvany [26, 27] propose that the stress functions should be replaced with functions approximating the stresses (smooth envelope functions) such that the permissible stress is increased when the cross-section areas become small. In Reference [28], it is proposed that the stress constraints and the lower bounds on the variables should be perturbed by a small positive parameter. It is shown in Reference [29] that the sequence of global optimal solutions to the perturbed problem converges to the set of global optimal solutions to the original problem as the parameter tends to zero. This approach is commonly called the ε-relaxed approach (unfortunately, the name is misleading since the perturbed problems are not relaxations of the original problem). The ε-relaxed approach is extended to topology optimization of continuum structures in References [30, 31] and to include local buckling constraints in Reference [20]. The use of the ε-relaxed approach and similar methods will in general not produce a global minimizer since the perturbed problems are intrinsically non-convex with several local minima, see Reference [24]. Furthermore, because of the positive lower bounds imposed on the area variables, the ε-relaxed approach will fail to recognize the favourable structure of the problem in the single load case. For a historical perspective on the research regarding singular topologies we refer to the review article Rozvany [27]. For an introduction to topology optimization of continuum and discrete structures, see e.g. References [32 35]. This paper is organized as follows. In Section 2, two equivalent formulations of the considered problem are presented and their properties are discussed. In Section 3, we present a continuous branch-and-bound algorithm for the considered truss topology optimization problem, derive valid inequalities, and establish convergence of the algorithm. In Section 4, we present an implementation of the proposed algorithm and computational results on a large set of problems both with and without buckling constraints. Section 5 contains conclusions and a discussion on further research. Section 6, finally, contains characteristics and obtained solutions for all problems Notations and basic relations The elastic equilibrium equations of a given truss structure with d degrees of freedom subject to an external load p R d are assumed to be given by K(x)u = p with K(x) = n x j (E j /l j )r j rj T Rd d (1) j=1 where K(x) R d d denotes the global stiffness matrix, u R d is the nodal displacement vector, n is the number of potential bars, and x = (x 1,...,x n ) T R n are the cross-section areas. Furthermore, E j > 0 is the Young s modulus, and l j is the length of the jth bar respectively. Moreover, r j R d contains the direction cosines such that rj T u is the linearized elongation of the jth bar. Let R denote the d n matrix whose jth column is r j. It is assumed that the rows of R are linearly independent. This is equivalent to the assumption that the stiffness matrix is positive definite if x>0. If a R then the real numbers a + and

4 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1273 a denote the positive and negative parts of a, i.e. a + = max{a,0} and a = min{a,0}. If a R n then a + R n and a R n are given component-wise, i.e. a + = (a + 1,...,a+ n )T and a = (a 1,...,a n )T.IfA and B are real symmetric matrices, the notation A B means that A B is positive semidefinite. 2. PROBLEM FORMULATIONS We consider the topology optimization problem of minimizing the weight of a truss structure subject to displacement, stress, and local Euler buckling constraints. Several equivalent formulations of this problem are available. In the nested formulation, only the cross-section areas are used as variables and the stresses and displacements are implicitly defined using the equilibrium equations (1). This formulation is considered in References [20, 28, 36]. The nested formulation is defined by the following non-convex problem: glob min x subject to n l j ρ j x j j=1 j σ jl (x) σ max j l if x j > 0 σ jl (x) σ cr j (x) l if x j > 0 l u l (x) l x min x x max l (2) where σ max j > 0 and j < 0 are given finite stress limits in tension and compression respectively, l and l are given finite displacement bounds satisfying l l, and x min and x max are given finite bounds on the areas satisfying 0 x min x max. ρ j denotes the density of the material used in the jth bar. The subscript l refers to the lth load condition. The objective function in (2) is the weight of the structure. The stress in the jth bar is given by σ jl (x) = (E j /l j )rj Tu l(x) where u l (x) is a solution to K(x)u l = p l.ifthejth bar is under compression the member stress must not exceed the Euler buckling stress given by σ cr j (x j ) = x j πe j 4l 2 j Here, and throughout, we assume that the cross-sections are circular. To avoid structures for which there does not exist a solution to the equilibrium equations, a small positive lower bound is normally imposed on the area variables. This ensures that the stiffness matrix is positive definite, but it destroys the combinatorial character of the problem. If a positive lower bound is imposed on the area variables, all stress and buckling constraints should be taken into consideration and the problem transforms into a sizing problem rather than a topology optimization problem. Alternatively, the topology optimization problem may be formulated in the cross-section areas, member forces, nodal displacements, and member stresses, see e.g. References [12, 16].

5 1274 M. STOLPE The formulation is defined by the following problem denoted by P: z=glob min x,f,u,σ n l j ρ j x j j=1 subject to Rf l = p l l σ jl = (E j /l j )r T j u l (j, l) f jl = x j σ jl (j, l) x j j f jl x j σ max j (j, l) f jl x j σ cr j (x j ) (j, l) cjl min jl cjl max (j, l) l u l l l x min x x max (3) where f jl R is the normal force in the jth bar, f l = (f 1l,...,f nl ) T R n, and the upper and lower bounds on the stress variables, denoted by cjl max and jl, are given by and c max jl = E j l j max rj T u l = E j ((r u l U j + l l )T l + (rj )T l ) j c min jl = E j l j min rj T u l = E j ((r u l U j + l l )T l + (rj )T l ) j where U l ={u l l u l l }. The constraints Rf l = p l correspond to node equilibrium of forces and the constraints f jl = x j σ jl = x j (E j /l j )rj Tu l correspond to geometric compatibility and Hooke s law. Together they imply that the equilibrium equations K(x)u l = p l are satisfied for all load conditions l. The linear stress constraints x j j f jl x j σ max j imply that the stress σ jl in the jth bar satisfies j σ jl σ max j when x j > 0, and that the member force is zero when the bar is not present. Hence, the stress and buckling constraints are effectively removed when the bar is not present. Therefore, the two formulations (2) and (3) are equivalent. Potentially, some of the bars in (2) and (3) may be forced to attain the same area, so-called variable linking. These conditions can be handled by eliminating redundant design variables. Here, we instead choose to preserve all the original variables and model variable linking conditions as linear equality constraints. Although equivalent, the properties of problems (2) and (3) are significantly different from a computational point of view. Since both the displacements and the forces are treated as variables in (3), the displacement and stress constraints may be modelled as linear constraints rather than non-convex constraints. Furthermore, the stiffness matrix need not be positive definite and hence there is no need to assign strictly positive lower bounds on the area variables. Another important difference between problem (2) and (3) is the representation of the feasible set. The set of

6 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1275 constraints in (2) depends on the current value of the area variables, while the set of constraints is fixed in (3). The non-linearities in (3) consist of mn bilinear equality constraints which constitute the compatibility equations and mn non-convex quadratic constraints which constitute the local buckling constraints. The well-defined structure of the non-linearities in (3) makes it possible to construct convex, in fact linear, relaxations of (3) which can be efficiently solved. 3. A BRANCH-AND-BOUND ALGORITHM In branch-and-bound algorithms the feasible set of the original problem is partitioned into subsets (branching). These subsets are then subsequently also split into subsets to refine the partition of the feasible set. For each subset a lower and optionally also an upper bound on the objective function value are determined (bounding). Typically, the lower bounding problems are convex problems that can be efficiently solved to global optimality. Upper bounds are given by the objective function value of points feasible to the original problem. One of the features of branch-and-bound is the ability to delete the subsets (fathoming) for which it can be proved that the optimum is not attained. This can be achieved by proving that the lower bounding problem over the subset is infeasible or by showing that the lower bound over the subset is larger than a known upper bound The algorithm Let F denote the feasible set of the non-convex problem (3) and let {F k : k = 1,...,t} be a partition of F, i.e. t k=1 F k = F and F k F i = F k F i for i, k = 1,...,t and i = k, where F k denotes the boundary of the set F k. Let P k denote the problem z k =glob min x,f,u,σ subject to n l j ρ j x j j=1 (x,f,u,σ) F k In our approach, each problem P k is of the same type as the original problem but with tighter bounds on the variables, obtained by rectangular partitioning. It follows that z = min 1 k t z k. Furthermore, let P R k denote a relaxation of P k with feasible set F R k F k. A branch-andbound method for solving (3) is given in Algorithm 3.1. In the algorithm, A is a collection of non-convex problems {P k }. Associated with each problem in A is a lower bound L k Lower bounding problem To construct a convex underestimating problem which is a relaxation to (3), we follow the practice used in e.g. References [37 39], and replace every bilinear term with its convex envelope. The convex envelope of a function over a closed convex set is defined as the pointwise supremum of all convex functions which underestimate the function over the set. We denote the convex envelope of a function g by co g. Let Ω denote a box in R 2 defined by Ω ={(ξ, η) R 2 ξ min ξ ξ max, η min η η max }

7 1276 M. STOLPE Lemma 3.1 The convex envelope of ξη over Ω is co ξη = max{ξ min η + η min ξ ξ min η min, ξ max η + η max ξ ξ max η max } For a proof, see Reference [40]. Lemma 3.2 co ξη = ξη for all (ξ, η) on the boundary of Ω. Can be proven by direct computations. Algorithm 3.1. (Framework for the branch-and-bound method) Initialize the best upper and lower bound U =+, and L =. Initialize the collection with active problems A ={P} with the lower bound L 0 =. while A = do (Node selection) Choose a problem P k from A. We choose the problem with the least lower bound. Delete the problem from the list of active problems, i.e. let A A\P k. (Lower bounding) Solve the relaxation P R k. Let L k denote the optimal value of the relaxation. If the relaxation is infeasible or if L k U then return to node selection. If the optimal solution to the relaxation is feasible (and thus optimal) to P k and the objective function value is better than the best upper bound, update U and return to node selection. Otherwise, proceed to upper bounding. (Upper bounding) Solve the optional upper bounding problem corresponding to P k using local minimization. Let U k denote the upper bound, and update the best upper bound U if U k U. (Partitioning) Partition F k into F k1, F k2,... and create a new set of problems P k1, P k2,..., and add them to A together with the lower bound L k. We use rectangular partitioning. (Fathoming) Prune A of problems with lower bounds greater than the best upper bound, i.e. A A\P k for all P k with L k U. Update the overall lower bound, i.e. let L = min i: Pi A{L i }. end while Lemma 3.3 The maximum separation of the bilinear term ξη from its convex envelope inside the rectangle Ω occurs at the midpoint ξ = 2 1 (ξmax + ξ min ), η = 2 1 (ηmax + η min ) and is equal to one-fourth 1 of the area of the domain, 4 (ξmax ξ min )(η max η min ). For a proof, see Reference [41].

8 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1277 The number of bilinear terms in the equations f jl = x j (E j /l j )r T j u l is equal to the number of non-zeros in the vector r j. The total number of bilinear terms may therefore be as large as 4mn for planar trusses and 6mn for three-dimensional trusses. Because of the additional variables σ jl representing the stress in each member, the number of bilinear terms can always be reduced to mn. The compatibility equations are given by f jl = x j σ jl, or equivalently by x j σ jl f jl and x j σ jl f jl. Replacing the bilinear terms x j σ jl and x j σ jl by their respective convex envelopes, the following linear constraints approximating the compatibility equations are obtained. f jl xj min σ jl + x j cjl min f jl xj max σ jl + x j cjl max f jl xj min σ jl + x j cjl max f jl xj max σ jl + x j cjl min xj min cjl min xj max cjl max xj min cjl max xj max cjl min To obtain convex constraints approximating the stability constraints we introduce additional variables w j representing xj 2 by adding the constraints x2 j w j and xj 2 w j. The stability constraints are then given by f jl σ cr j (w j ). Replacing the quadratic terms xj 2 and xj 2 by their respective convex envelopes, the following linear constraints approximating the stability constraints are obtained. w j 2x min j w j 2x max j w j x min j f jl σ cr j (w j ) x j xj min xj min x j xj max xj max x j + x max j x j xj min xj max The following linear program is an underestimating problem for (3): z lp =minimize x,f,u,σ,w n l j ρ j x j j=1 subject to Rf l = p l l σ jl = (E j /l j )r T j u l (j, l) (4) (5) x j j f jl x j σ max j (j, l) (x, f, σ) satisfies (4) (j, l) (6) (x, f, w) satisfies (5) (j, l) cjl min jl cjl max (j, l) l u l l l x min x x max

9 1278 M. STOLPE where w = (w 1,...,w n ) T R n. Let F R denote the feasible set of the lower bounding problem (6). Since the feasible set of the non-convex problem (3) is contained in the feasible set of the linear program (6), i.e. F R F, it follows that (6) is a relaxation of (3) Generating valid inequalities It is possible to strengthen the lower bounding problem (6) by adding so-called valid inequalities (additional constraints) to the feasible set of relaxation (6). If U is a known upper bound on the optimal objective function value in (3), an inequality is called valid for (3) if it does not exclude any solutions of (3) with objective function values better than U. Hence, an inequality which is valid for (3) will not exclude any global optimal solutions of (3). A valid inequality may however remove feasible, but non-optimal, solutions to (3) and may therefore be used to tighten the feasible set of (3) and subsequently also the feasible set of relaxation (6). Here, we will show how convex minimum compliance problems and convex minimum weight problems can be used to generate valid inequalities Valid compliance inequalities. The compliance 1 2 pt l u l is a measure of the stiffness of the truss for a given load p l and corresponding displacement vector u l. Because of the equilibrium equations K(x)u l = p l, the compliance may be written 1 2 pt l u l = 1 2 ut l K(x)u l for all x such that there is a displacement vector u l satisfying the equilibrium equations. Since the stiffness matrix is assumed to be positive semidefinite p T l u l 0 for all (x,f,u) feasible to (3). Therefore, p T l u l 0 is a valid inequality for (3). This inequality may however not be satisfied in an optimal solution of the lower bounding problem (6). These cuts may be further strengthened by solving convex minimum compliance problems. Consider the following weighted minimum compliance problem, where α l R are given non-negative constants for all l. c avg =minimize x,u m α l pl T u l l=1 subject to K(x)u l = p l l n ρ j l j x j U j=1 x min x x max (7) where U is an upper bound for the minimum weight problem (3). Under mild assumptions, existence of a global minimizer to (7) is assured, see e.g. Reference [42]. Because of the equilibrium equations (7) is a non-convex problem. However, (7) may equivalently be written as a non-smooth convex problem, see e.g. Reference [42], or as the following program with a linear objective function and convex quadratic constraints, see e.g.

10 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1279 Reference [43]. maximize u,λ,s subject to 2 m α l pl T u l λu/ρ + n s j l=1 j=1 ( λ m l=1 ( λ m λ 0 l=1 α l u T l K j u l ) l j x min j s j j α l u T l K j u l ) l j x max j s j j (8) where s R n, λ R, and K j = (E j /lj 2)r j rj T. Here, we assume that the density ρ j is the same in all elements, i.e. ρ j = ρ for all j. Since U is an upper bound on the optimal value of the weight of the structure, the feasible set of (7) is a relaxation of the feasible set of the non-convex problem (3). Since the minimum compliance problem (7) may equivalently be written as a convex optimization problem, c avg is a lower bound on the weighted compliance for the optimal solution of the non-convex problem (3). We have thus proved the following. Proposition 3.1 If (u, λ,s ) is an optimal solution of (8) and U is an upper bound on the optimal objective function value in (3), then is a valid inequality for (3). m α l pl T u l m α l pl T u l (9) l=1 l=1 Alternatively, (7) may equivalently be written as the following linear semidefinite program, see e.g. References [44, 45] c avg =minimize x,τ subject to τ ( τ p T α p α K α (x) ) n ρ j l j x j U j=1 x min x x max 0 (10) where K α (x) R md md is given by K α (x) = diag(α 1 K(x),...,α m K(x)) and p α = (α 1 p T 1 α m p T m )T R md. Since the area variables are not eliminated in (10), variable linking constraints are easily incorporated when using this formulation. The following valid inequality is obtained.

11 1280 M. STOLPE Proposition 3.2 If (x, τ ) is an optimal solution of (10) and U is an upper bound on the optimal objective function value in (3), then is a valid inequality for (3). m α l pl T u l τ (11) l=1 If the lower bounds on the area variables are strictly positive, i.e. x min > 0, then the stiffness matrix is positive definite for all x such that x min x x max and the displacement variables may be eliminated from the non-convex minimum compliance (7). In this case (7) may instead be written as the following convex optimization problem on standard form: minimize x subject to m α l pl T K 1 (x)p l l=1 n ρ j l j x j U j=1 x min x x max (12) If the cuts are generated using the original lower and upper bounds on the area variables the cuts will be valid throughout the entire branch-and-bound tree. These cuts may then be strengthened every time the global upper bound U is improved. Cuts may also be generated at every node in the search tree using the lower and upper bounds on the area variables available at the node. These cuts will not be valid in the entire tree. This is computationally expensive, but may strengthen the relaxations and therefore decrease the size of the branch-and-bound tree Valid area and weight inequalities. Since, the displacement variables are assumed to be bounded it is possible to obtain an upper bound on the compliance for each load condition by maximizing the compliance over the feasible set of the linear relaxation (6), i.e. by solving the following linear program: cl max =maximize x,f,u,σ,w subject to pl T u l (x,f,u,σ,w) F R n l j ρ j x j U j=1 (13) where F R denotes the feasible set of the linear relaxation (6). Since U is an upper bound on the optimal value of the weight of the structure, the feasible set of (13) is a relaxation of the feasible set of the non-convex problem (3). Since (13) is a linear program, it follows that cl max is an upper bound on the compliance for load condition l for any optimal solution of (3). Therefore, the following proposition holds.

12 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1281 Proposition 3.3 If (x,f,u, σ,w ) is an optimal solution of (13) and U is an upper bound on the optimal objective function value in (3), then is a valid inequality for (3). p T l u l p T l u l This approach may seem counter-intuitive but it provides the possibility to improve the bounds on the area variables by solving a sequence of convex optimization problems. Consider the following minimum area problem with compliance constraints which is obtained if the stress, displacement, and local buckling constraints are relaxed in (3). minimize x,u x j subject to K(x)u l = p l l pl T u l cl max l L n ρ j l j x j U j=1 x min x x max (14) where L is a lower bound on the optimal objective function value of (3). Because of the equilibrium equations, (14) is a non-convex problem. Fortunately, (14) may equivalently be written as the following linear semidefinite program, see e.g. Vandenberghe and Boyd [44] and Ben-Tal and Nemirovski [45]. minimize x subject to x j ( c max l p l p T l K(x) ) L n ρ j l j x j U j=1 x min x x max 0 l (15) Since pl Tu l cl max for all l are valid inequalities to (3) the feasible set of (15) is a convex relaxation of the feasible set of (3). Therefore, the following proposition holds. Proposition 3.4 If x is an optimal solution of (15) and L and U are lower and upper bounds on the optimal objective function value in (3), respectively, then is a valid inequality for (3). x j x j (16)

13 1282 M. STOLPE If the minimization in (15) is replaced by maximization x j xj is a valid inequality for (3). If the lower bounds on the area variables are strictly positive, i.e. x min > 0, then the stiffness matrix is positive definite for all x such that x min x x max and the displacement variables may be eliminated from the non-convex problem (14). In this case (14) may instead be written as the following convex optimization problem on standard form: minimize x j x subject to pl T K 1 (x)p l cl max L n ρ j l j x j U j=1 x min x x max This approach also enables us to find another lower bound on the optimal objective function value in (3) by solving the following linear semidefinite program: n z sdp =minimize ρ x j l j x j subject to j=1 ( c max l p l p T l K(x) ) x min x x max l 0 l The lower bound obtained by solving (18) may potentially be much better than the bound obtained when solving (6). Proposition 3.5 If x is an optimal solution of (18) then n ρ j l j x j n is a valid inequality for (3). j=1 j=1 (17) (18) ρ j l j x j (19) 3.4. Upper bounding In order to achieve good upper bounds on the optimal value, local minimization is performed on problem (3) at every rth node in the branch-and-bound tree using the optimal solution of the lower bounding problem (6) as starting point. The local minimization is done using a sequential quadratic programming package which exploits the sparsity in the linear constraints as well as the sparsity in the Jacobian of the non-linear constraints Node selection The node chosen for further refinement is the one where the least lower bound is attained (the best-bound-first rule). This node selection rule is, by definition, so called bound improving, see Reference [8].

14 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B Partitioning The feasible domain at the current node is partitioned into a finite number of subregions through rectangular partitioning. The branching variable is selected as the one which (in some sense) contributes the most to the gap between the upper and lower bounds. Since, the only variables that appear in non-convex terms are the area variables x j and the stress variables σ jl these are the only variables considered for branching. We will throughout denote them non-convex variables. The branching variable and branching point are chosen as follows. Every 25th iteration the non-convex variable with the largest distance between its upper and lower bounds is branched at the midpoint (bisection). In all other iterations, we compute the violations of the nonlinear constraints obtained with the solution of the lower bounding problem. The violation of the compatibility equations is defined to be the actual violation in the optimal solution to the lower bounding problem divided by the maximal separation of the bilinear term from its convex envelope, see Lemma 3.3, e jl = (x max j f lp jl xlp j σlp xj min jl )(cjl max c min jl ) where (x lp,f lp,u lp, σ lp,w lp ) is the optimal solution of the linear programming relaxation (6). Similarly, the violation of the buckling constraints is defined to be ej b = wlp j xlp j xlp (xj max xj min ) 2 From the variables responsible for the largest violations, we choose the one with the largest relative distance from the solution of the lower bounding problem to the nearest bound as branching variable. The branching value is now given as follows. If the incumbent is feasible with respect to the variable bounds at the current node the value of the incumbent is used as branching point. Otherwise, the branching point is given as the solution to the lower bounding problem. If f jl and σ jl have different signs in the solution of the lower bounding problem, the following constraints are valid inequalities and are used as both branching points and branching variables. It must hold that either f jl 0, σ jl 0orf jl 0, σ jl 0. We call this procedure disjunctive branching. This approach does not produce a partition since the subsets f jl > 0, σ jl < 0 and f jl < 0, σ jl > 0 are not included. However, these subsets are infeasible with respect to the original problem and can therefore immediately be deleted from the search tree Solving the lower bounding problem To accelerate the convergence of the branch-and-bound method we apply range reduction techniques at each node of the search tree. Prior to solving the lower bounding problem, the variable bounds are tightened using feasibility based range reductions (preprocessing), see e.g. Reference [46]. To tighten the variable bounds every non-convex variable is minimized as well as maximized over the feasible set of the lower bounding problem (6), see e.g. Reference [7]. If at least one bound is improved by a prespecified positive amount ε v red > 0 the lower bounding problem is reconstructed and the procedure is repeated. The maximum number of j

15 1284 M. STOLPE iterations in this procedure is 5 if n 10 and 3 otherwise. This procedure is computationally expensive but it may decrease the size of the search tree considerably. After the lower bounding problem is solved the variable bounds may be tightened using optimality based range reductions (postprocessing), see e.g. Reference [3]. Valid inequalities improving the bounds are derived from both active and inactive variable bound constraints. When deriving valid inequalities from inactive bound constraints the variable under consideration is temporarily fixed to its lower or upper bound and the lower bounding problem is again solved (probing). Probing is performed only on non-convex variables. The procedure for solving the lower bounding problem is shown in Algorithm 3.2. Algorithm 3.2. (Framework for solving the lower bounding problem) (Preprocessing) Tighten the variable bounds on the relaxation P R k using feasibility based range reductions. (Cut generation) Generate valid inequalities and add them to the P R k. (Solving) Solve the relaxation P R k. Let L k denote the optimal value of the relaxation. If the relaxation is infeasible or if L k U then return to node selection. If the optimal solution to the relaxation is feasible to P k and the objective function value is better than the best upper bound, update U and return to node selection. (Postprocessing) Tighten variable bounds on the relaxation P R k using optimality based range reductions. If at least one bound is improved by a prespecified amount ε v red > 0, then reconstruct the relaxation and return to the preprocessing stage. Valid compliance inequalities (9) are generated at the root node and thereafter at all active nodes on every 5th level of the search tree. One inequality per load condition and one inequality involving all load conditions are generated. If no variable linking is imposed the minimum compliance problem (7) is modelled and solved as the convex problem (8). If, on the other hand, variable linking is present, (7) is modelled and solved as the semidefinite program (10) or as the convex program (12). Valid area inequalities of type (16) improving both the lower and upper bounds on all the area variables are also generated at the root node and thereafter at all active nodes on every 5th level of the search tree Convergence of the algorithm Convergence of the branch-and-bound algorithm is guaranteed by the following observations, see Reference [8, Theorem IV.3]. (i) The node selection rule is, by definition, bound improving. (ii) The bounding operation is consistent, see Reference [3]. (iii) Algorithm 3.2 does not cycle. This follows from the compactness of the feasible set of the lower bounding problem. In practice, the lower bounding algorithm is terminated when the improvement in the objective function is smaller than a certain predefined positive amount or when the number of iterations between the postprocessing phase and the preprocessing phase exceeds a prespecified positive integer.

16 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1285 (iv) The feasibility and optimality based range reductions are valid, see Reference [3]. (v) The compliance inequalities (9) and (11) and the area inequalities (16) and (19) are valid. 4. AN IMPLEMENTATION OF THE METHOD The branch-and-bound method has been implemented in MATLAB [47]. The MATLAB code controls the overall program flow and the branch-and-bound tree. All linear programs are constructed using MATLAB and solved using the fast and robust simplex solvers in CPLEX 8.0 [48]. CPLEX is called from MATLAB using a MEX interface, which allows problem modifications such that CPLEX can take advantage of the current solution when resolving a problem after slight modifications. The upper bounding problems are solved using the sequential quadratic programming package SNOPT, see Gill et al. [49, 50]. Again, a MATLAB MEX interface is used to access SNOPT. SNOPT does callback to MATLAB to compute the objective function and the constraints as well as the gradient of the objective function and the Jacobian of the constraints at the current iteration point. The convex minimum compliance problems (8), (12), and the convex minimum weight problem (17) are solved with SNOPT. The semidefinite programs (10), (15), and (18) are solved with the MATLAB toolbox SeDuMi, see Reference [51]. If the lower bounds on the area variables are strictly positive, then the minimum weight and minimum compliance problems are modelled and solved as standard convex problems rather than semidefinite programs. All computations are performed on an Intel Pentium 4 (2.0GHz clock frequency and 256Mb internal memory) PC running Linux RedHat Feasibility, optimality, and parameters A point (x,f,u) is considered to be feasible with respect to the non-linear constraints in (3) if f jl x j (E j /l j )rj Tu l ε feas and f jl x j σ cr j (x j ) ε feas where ε feas > 0 is the feasibility tolerance. A point (x,f,u) is considered to be feasible with respect to the linear equality constraints in (3) if Rf l p l / p l ε feas, and feasible with respect to the inequality constraints ε feas + x j j f jl x j σ max j + ε feas. A solution is considered feasible with respect to the variable bounds if they are not violated by more than ε feas. The algorithm is stopped if the list of active nodes becomes empty or if the relative gap (U L)/L is less than the optimality tolerance ε rel > 0 or if the absolute gap U L is less than the optimality tolerance ε abs > 0. A node is considered to be fathomed if L k >U ε opt. The following tolerances are used ε opt = 10 7, ε feas = 10 7, ε rel = 10 4, ε abs = 10 4, and ε v red = The feasibility and optimality tolerances are set to 10 9 for CPLEX and 10 8 for SNOPT. For all other options default values were used. For SeDuMi default tolerances and options were used. In Algorithm 3.2 at most 5 iterations are allowed between the preprocessing and postprocessing phases. For problems with no more than 5 potential bars the upper bounding problem is solved at every node in the search tree. For problems with 5 <n 10 bars, the upper bounding problem is solved at every 10th node. For problems with n>10 the upper bounding problem is solved at every 100th node.

17 1286 M. STOLPE 4.2. Detecting favourable situations To detect the favourable situation that the stress and displacement constrained program (3) is equivalent to a linear program the following steps are performed before entering the branch-andbound algorithm. First, the linear lower bounding problem (6) is solved. The optimal solution is denoted by (x lp,f lp,u lp, σ lp,w lp ). If the point (x lp,f lp,u lp ) is feasible to the original non-convex problem (3) the solution is also optimal to (3) and the algorithm is terminated. Otherwise, an attempt is made to construct a displacement vector ū which together with the area and force variables obtained in the solution to the linear relaxation (6)) satisfies the displacement bounds as well as the compatibility equations f lp jl = xlp j (E j /l j )rj Tūl. To this end, the following minimum norm problem is solved: minimize u subject to lp max{ fl DRT u l } l l u l l where D R n n is the diagonal matrix D = diag(x lp 1 E 1/l 1,...,xn lp E n /l n ). Problem (20) can equivalently be cast as a linear program. Denote the optimal solution to (20) by ū. If the optimal value of (20) is zero the algorithm is terminated and a feasible, and hence optimal, solution to (3) is given by (x lp,f lp, ū). Otherwise, a final attempt to solve problem (3) without using branch-and-bound is made. It is possible that the semidefinite lower bounding problem (18) may provide a better lower bound than the linear relaxation (6). Therefore, the lower bounding problem (18) is solved. If the lower bound is improved, then the upper bounding problem is also solved and the relative and absolute gaps are computed. If either the relative or absolute gap is sufficiently small the solution obtained when solving the upper bounding problem is optimal, otherwise the branch-and-bound algorithm is called Computational results A collection of 33 stress and displacement constrained problems taken from the literature were solved to global optimality. The number of potential bars in the test set range from 2 to 25 and hence the problems are from a truss topology optimization point of view small-scale. However, for the considered problem of finding a global optimal solution the largest problem in the test set should be considered as large-scale. For most of these problems, computational results are available in the literature. For some of the problems global optimality has not previously been proved. Results concerning problem sizes, optimal weights, number of iterations, and CPU-time are shown in Table I. Most of the problems in the test set originally contain only stress and displacement constraints and become infeasible when adding local buckling constraints. Buckling constraints are therefore only included in the problems numbered 5, 27, 28, and 33. Computational results for this reduced set of problems with buckling constraints are shown in Table II. Problem definitions as well as the obtained optimal designs can be found in Section 6. In the tables, N, N o, and N m denote the total number of nodes in the search tree, the node in which the optimal solution was found, and the maximum number of active nodes, respectively. T a denotes the time spent on generating valid area inequalities, i.e. constructing and solving the minimum weight problems (15), (17), and (18). T c denotes the time spent on generating valid compliance inequalities, i.e. constructing and solving the minimum compliance l (20)

18 GLOBAL OPTIMIZATION OF TRUSS TOPOLOGY USING B & B 1287 Table I. Computational results for problems with stress and displacement constraints. Problem Tree size CPU (s) Gap (%) Objective value No. References n m d N N m N o T a T ub T c T pre T post T b&b T ν i (%) z lp z sdp L z 1 [52] [28] [24] [24] [53] [22] [54] [55] [55] [55] [55] [55] [55] [55] [55] [55] [55] [28] [28] [27] [21] [21] [22] [21] [21] [21] [20] [20] [55] [55] [55] [55] [20]

19 1288 M. STOLPE Table II. Computational results for problems with stress, displacement, and local buckling constraints. Problem Tree size CPU (s) No. References n m d N N m N o T a T ub T c T pre T post T b&b T 5 [20] [20] [20] [20] Gap (%) Objective value ν i (%) z lp z sdp L z problems (8), (12), and (10). T b&b denotes the time for maintaining the branch-and-bound tree, branching, feasibility control, and constructing and solving the linear relaxation (6). T ub denotes the time spent on solving the upper bounding problems. T pre denotes the time used for feasibility based range reductions while T post denotes the time for optimality based range reductions. T denotes the overall CPU-time used to solve the problem. Further, z denotes the obtained weight of the best feasible solution found, z lp denotes the initial lower bound obtained by solving the linear lower bounding problem (6), and z sdp denotes the initial lower bound obtained by solving the linear semidefinite minimum weight problem (18). The initial relative gap ν i is defined as ν i = (z z lp )/z lp Comments on the computational results We begin with the situation where no buckling constraints are included. In this case and due to the small number of bars, problems 1, 3, 4, 7, 20, 21, and 22 have known solutions. These solutions were all verified using branch-and-bound. Problems 6, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 27, 29, and 31 were all solved without solving the upper bounding problem or entering branch-and-bound. These problems are therefore essentially equivalent to linear programs and are therefore not suitable as benchmark examples. For problems 2, 5, 8, 16, 24, 25, 26, 28, 30, 32, and 33 global optimality of previously known feasible solutions was verified. Problems 7, 9, 13, and 17 are, as far as we know, solved to global optimality for the first time. For problem 12 the obtained solution improved on the previously known solution. Now to the situation where buckling constraints are also included. Global optimality of previously known solutions for problems 28 and 33 was verified. For problems 5 and 27 the obtained solution improved on the previously known solutions given in Reference [20]. For problem 27 global optimality of the solution given in [56] was proven. Problems 9 and 17 are identical to problems 8 and 16, respectively, with the important exception that the strictly positive lower bounds on the design variables are replaced by zero. This is the reason why the time used to generate valid area inequalities is significantly larger for problems 9 and 17 than for problems 8 and 16, respectively, even though the number of nodes in the search tree is smaller. For problems 9 and 17, the area inequalities are generated