BGE Research Report No. 04-05 Building Geoenvironment Engineering Laboratory Department of Urban and Environmental Engineering Kyoto University Sakyo, Kyoto 606-8501, Japan
Sequential Semidefinite Program for Robust Truss Optimization based on Robustness Functions associated with Stress Constraints Y. Kanno† and I. Takewaki‡ July 2004
Abstract A robust truss optimization scheme, as well as an optimization algorithm, is presented based on the robustness function. Under the uncertainties of external forces based on the info-gap model, the maximization problem of robustness function is formulated as the optimization problem with infinitely many constraint conditions. By using a semidefinite relaxation technique, we reformulate the present problem to a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has the global convergent property. It is shown, in numerical examples, that optimum designs of various trusses can be found without any difficulty. Keywords: robust optimization; info-gap model; semidefinite program; structural optimization; successive linearization method
† ‡
Department of Urban and Environmental Engineering, Kyoto University e-mail:
[email protected] Department of Urban and Environmental Engineering, Kyoto University e-mail:
[email protected]
1. Introduction. Let f : Rn → Rkf be a convex quadratic function. For a continuously differentiable function g : Rm → Rkh , define the set Q by Q = {Rm |g(q) ≤ 0}. The set consists of all subsets of a set X is denoted by P(X ). W : Rm+1 → P(Rn ) denotes the point-to-set mapping satisfying W(s1 , q 1 ) ⊂ W(s2 , q 1 ) if s1 < s2 . We make further assumptions on W (see Section 3). For the fixed parameters q ∈ Q, consider the following parametric semi-infinite programming problem in variables (s, w) ∈ R × Rn : max s (1) s.t. f (w) ≤ 0 ∀w ∈ W(s, q). The optimal value function β : Q → [−∞, +∞] of Problem (1) is defined as β(q) = sup {s : f (w) ≤ 0 ∀w ∈ W(s, q)} .
(2)
This paper deals with the maximization problem of β(q) over Q, which is alternatively formulated as the following optimization problem in variables (t, w, s) ∈ R × Rn × Rm : ⎫ ⎪ max s ⎬ (3) s.t. f (w) ≤ 0 ∀w ∈ W(s, q), ⎪ ⎭ g(q) ≤ 0. Problem (3) is motivated by the novel concept of robust truss optimization under stress constraints, which is based on the robustness function proposed by Ben-Haim [1]. A truss is an assemblage of nodes connected by members, or bars, that transmit only axial forces. In the context of robust truss optimization, the variables q in Problem (3) denote the so-called design variables of the truss, e.g., member cross-sectional areas of the truss. w is regarded as the vector of state variables such as the nodal displacements. The constraints f (w) ≤ 0 correspond to the constraints on the performance of truss, e.g., stress constraints of members. The design variables q should also have some constraints expressed as g(q) ≤ 0, e.g., the nonnegativity of member cross-sectional areas. Then the robustness function of the truss with the specified design q is given as β(q) in (2). We attempt to find the truss design which maximizes the robustness function. This structural optimization problem is referred to as the maximization problem of robustness function, which is shown to be formulated in the form of Problem (3). The major difficulty in the maximization problem (3) of robustness function is that Problem (3) has the infinitely many constraints. To overcome this difficulty, we first show in Section 3 that Problem (1) can be reformulated into the semidefinite programming (SDP ) problem [2]. Then, in Section 4, Problem (3) is shown to be equivalent to the so-called nonlinear SDP problem [3], namely, the linear optimization with the constraints such that the matrix defined as the nonlinear function of some variables should be positive semidefinite. Based on the successive linearization method [4, 5], we propose an algorithm with the global convergence property under certain assumptions, at each iteration of which we solve the (linear) SDP problems by using the primal-dual interior-point method [2, 6]. The robust optimal design has received increasing attention, which may yield a system response that is less sensitive to uncertainties of parameters of the system [7–17]. Based on the stochastic 1
uncertainty model of mechanical parameters, various methods were proposed for reliability-based optimization. The structural optimization by minimizing the failure probability was studied, where the failure probability was estimated by using the Monte Carlo simulation [7, 8] with the responsesurface approximation [9]. In order to reduce the computational cost in the evaluation of the failure probability, the reliability index approach was utilized [10, 11]. Various formulations for sensitivity analysis of probabilistic constraints were also proposed [12, 13]. Doltsinis and Kang [14] performed the multi-objective optimization so as to minimize both the expected value and the standard deviation of the goal performance. On the other hand, as a non-probabilistic but bounded uncertainty model, the so-called convex model has been established [18]. A unified methodology of robust counterpart of the various convex optimization problems was developed by Ben-Tal and Nemirovski [19]. Calafiore and El Ghaoui [20] proposed a method for finding the ellipsoidal bound of the solution set of uncertain linear equations based on the semidefinite programming relaxation. For further application of the robust optimization, see Ben-Tal and Nemirovski [15] and Wolkowicz et al. [2, Ch. 6]. In the field of structural optimization, Pantelides and Ganzerli [16] proposed a truss optimization method by using the convex model for uncertainties. Han and Kwak [17] attempted to find the design which minimizes the infinity norm of a vector of sensitivity coefficients of the performance functions. Recently, based on the info-gap decision theory, the concept of robustness function was proposed by Ben-Haim [1]. The robustness function expresses the greatest level of non-probabilistic uncertainty at which any constraint in a mechanical system cannot be violated. The robustness function has the advantage, compared with the reliability analyses based on a stochastic uncertainty model, such that engineers do not have to estimate neither the level of uncertainty nor the probabilistic distribution of uncertain parameters, i.e., the robustness function does not require any information on statistical variation of the parameters of mechanical system, which is often difficult to obtain practically. Recently, the authors proposed the computable formulations of robustness functions for the given truss designs [21]. However, to the authors’ knowledge, neither algorithm nor methodology has been presented for robust structural optimization based on the robustness functions. This paper is organized as follows. In Section 2, in order to make this paper self-contained, we formulate the robustness function of the truss associated with stress constraints in the form of Problem (1). To help the readers to understand the concept of robustness functions, the illustrative truss example was given by Kanno and Takewaki [21, § 3]. We also briefly introduce SDP [2] and some technical lemmas on the linear matrix inequality. In Section 3, we show that the robustness function of the given truss design is obtained by solving the SDP problem. Namely, we reformulate Problem (1) into the SDP problem. Section 4 gives the rigorous definition of the maximization problem of robustness function, which is formulated as the nonlinear SDP problem. An algorithm based on the successive linearization method is also proposed. Section 5 investigates the global convergent behavior of our algorithm. In Section 6, the optimal designs of various trusses are found in order to see the effectiveness of the proposed method in view of computational efficiency and accuracy of the solutions. 2. Preliminaries. Throughout the paper, all vectors are assumed to be column vectors. However, for vectors
2
p ∈ Rn and q ∈ Rm , we often write (p, q) = (p , q ) ∈ Rn+m in order to simplify the notation. The standard Euclidean norm p2 = (p p)1/2 of a vector p ∈ Rn is often abbreviated by p. n ⊂ S n and S n ⊂ S n , Let S n ⊂ Rn×n denote the set of all n × n real symmetric matrices. S+ ++ + respectively, denote the sets of all positive semidefinite and positive definite matrices. We write n and P − Q ∈ S n . P O and P Q, respectively, if P ∈ S+ + 2.1. Robustness function associated with stress constraints. In this section, we show that the robustness function of trusses associated with the stress constraints is defined as the optimal objective value of a mathematical programming problem with infinitely many constraint conditions. For the robustness function of a simple illustrative truss, see Kanno and Takewaki [21, § 3]. Consider a linear elastic truss in the three-dimensional space. Let nd denote the number of d d degrees of freedom of displacements. u ∈ Rn and f ∈ Rn denote the vectors of nodal displacements and external forces, respectively. The system of equilibrium equations can be written as Ku = f ,
(4)
where K ∈ S n denotes the stiffness matrix of the truss. Let a = (ai ) ∈ Rn denote the vector of cross-sectional areas, where nm denotes the number of members. It follows from rank (∂K/∂ai ) = 1 (i = 1, . . . , nm ) that K can be written in a form of d
m
n m
K(a) =
i=1
ai bi b i .
(5)
Here, for each i = 1, . . . , nm , bi = (bij ) ∈ Rn is a constant vector. Let E denote the elastic modulus. It follows from (5) that the stress constraints are written as √ c u (i = 1, . . . , nm ), (6) Eb i ≤ σi d
where R σic > 0 denote the specified admissible stress of the ith member. However, instead of (6), we deal with the constraints in a slightly more general form of c |d l u − vl | ≤ ul
(l = 1, . . . , nc ),
(7)
where Rn uc = (ucl ) ≥ 0, dl ∈ Rn , and vl ∈ R are constant. Throughout the paper, we restrict ourselves to the cases where only the external forces f in (4) and α ≥ 0, we denote ∈ Rnd denote the nominal value of f . For the given f have uncertainty. Let f the uncertainty set of f by c
d
d T (α, f ) ⊂ Rn . ζ Suppose that T (α, f ) is given as the affine image of a set Z(α) := {ζ} ⊂ Rn satisfying the following assumptions:
3
Assumption 2.1. (i) Z(0) = {0}; (ii) If 0 ≤ α1 < α2 , then Z(α1 ) ⊂ Z(α2 ). Assumption 2.1 is motivated by the axioms of info-gap uncertainty model proposed by Ben-Haim d [1]. In what follows, T (α, f ) is often abbreviated by T (α) or T . Let U (α, a) ⊆ Rn denote the set of all the possible solutions to (4), i.e.,
nd (8) U (α, a) = u ∈ R K(a)u = f , for some f ∈ T (α) . Consider the following semi-infinite programming problem: c α∗ = max α : ucl ≥ |d l u − vl | (l = 1, . . . , n )
∀u ∈ U(α, a) .
(9)
The robustness function α : Rn × Rn → (−∞, +∞] associated with the stress constraints (7) is defined as (see, e.g., [1, Ch. 3]) ⎧ ⎨α∗ (if Problem (9) is feasible), α (a, uc ) = ⎩0 (if Problem (9) is infeasible). m
c
. For the two different vectors of cross-sectional In what follows, α (a, uc ) is often abbreviated by α m m (a1 , uc ) > α (a2 , uc ). If areas a1 ∈ Rn and a2 ∈ Rn , we say that a1 is more robust than a2 if α 1 α(a1 , uc )) satisfies ζ ∈ Z( ∃l ∈ {1, . . . , nc }
s.t.
ucl = |d l u − vl |,
then we say that ζ 1 is the worst case. Consequently, the robustness function α can be obtained by solving the optimization problem (9). However, it should be emphasized that Problem (9) is numerically intractable, because it has infinitely many constraints. This motivates us to investigate in Section 3 the SDP formulation of Problem (9). 2.2. Semidefinite program. For any P ∈ S n and Q ∈ S n , P • Q denotes the standard inner product of P and Q in the linear space S n ; i.e., P • Q = tr(P Q) = ni=1 nj=1 Pij Qij . The semidefinite programming (SDP ) problem refers to the optimization problem having the form of [2] (PSDP ) : min C • X (10) s.t. Ai • X = bi (i = 1, . . . , m), S n X O, where X is the variable, and Ai ∈ S n (i = 1, . . . , m), b = (bi ) ∈ Rm , and C ∈ S n are constant. The dual of Problem (10) is formulated in variables y ∈ Rm as ⎫ ⎪ (DSDP ) : max b y ⎬ m (11) Ai yi O. ⎪ s.t. C − ⎭ i=1
4
SDP has received increasing attention for its wide fields of application [6, 22, 23]. The primaldual interior-point methods, which were developed for the linear program [24] at first, have been naturally extended to SDP [25]. It is theoretically guaranteed that the primal-dual interior-point method converges to the optimal solutions of the pair of SDP problems (PSDP ) and (DSDP ) within the number of arithmetic operations bounded by a polynomial of n and m [6, 25]. 2.3. Technical lemmas. The reminder of this section is devoted to introducing some technical results that will be used in the following sections. Lemma 2.2 (Homogenization). Let Q ∈ S n , p ∈ Rn , and r ∈ R. Then ξ Q p ξ ≥0 r 1 p 1
Q p O. p r
∀ξ ∈ Rn
⇐⇒
Proof. See Lemma A.3 in Calafiore and El Ghaoui [20]. Lemma 2.3 (S-Lemma). For Qi ∈ S n , pi ∈ Rn , and ri ∈ R (i = 0, 1), define fi : Rn → R as fi (ξ) = ξ Qi ξ + 2p i ξ + ri . The implication f1 (ξ) ≥ 0
=⇒
f0 (ξ) ≥ 0
holds if and only if there exists a τ ≥ 0 such that f0 (ξ) − τ f1 (ξ) ≥ 0
∀ξ ∈ Rn .
Proof. See Lemma A.4 in Calafiore and El Ghaoui [20]. Lemma 2.4 (Lemma on the Schur complement). Let P A X= A Q n and Q ∈ S m . Then X O if and only if Q − be a symmetric matrix with blocks P ∈ S++ AP −1 A O.
Proof. See Lemma 4.2.1 in Ben-Tal and Nemirovski [6].
3. SDP formulation of robustness function. d Let F ⊂ Rn denote the set of all displacement vectors satisfying the stress constraints (7), i.e.,
d c u − v | (l = 1, . . . , n ) . (12) F = u ∈ Rn ucl ≥ |d l l Observing that ucl ≥ |d l u − vl | 5
is equivalent to 2 (ucl )2 − (d l u − vl ) ≥ 0,
we see that (12) is reduced to ⎫ ⎧ ⎬ ⎨ vl dl −dl dl u u d c ) . ≥ 0 (l = 1, . . . , n F = u ∈ Rn ⎭ ⎩ vl d (ucl )2 − vl2 1 1 l
(13)
, i.e., Throughout this section, for the given a, we assume that u ∈ F if u satisfies K(a)u = f Problem (9) has the nonempty feasible set. The following result shows that, under some assumptions on the uncertainty set, the robustness function α can be obtained by solving an SDP problem: Proposition 3.1. Assume that T and Z are given by
T (α, f ) = f + ζ ζ ∈ Z(α) ,
nd Z(α) = ζ ∈ R α ≥ ζ2 .
(14) (15)
For t ∈ R, ρ = (ρl ) ∈ Rn , and a ∈ Rn , define Gl (t, ρ, a) ∈ S n +1 (l = 1, . . . , nc ) by −ρl dl d ρl vl f l − K(a)f l + K(a)K(a) . Gl (t, ρ, a) = K(a) f ρl [(uc )2 − v 2 ] − t + f ρl vl f − f c
m
d
l
l
(16)
l
(a, uc ) is obtained by solving Then, for the given a ∈ Rn and uc ∈ Rn , the robustness function α c the following SDP problem formulated in variables (t, ρ) ∈ R × Rn : m
c
α (a, uc )2 = max {t : Gl (t, ρ, a) O,
ρl ≥ 0
(l = 1, . . . , nc )} .
(17)
Proof. We simply write K = K(a) till the end of the proof. In accordance with (14), U defined by (8) is rewritten as
d = ζ, for some ζ ∈ Z(α) . (18) U = u ∈ Rn Ku − f By using (15), we see that u ∈ U if and only if ) ≤ α2 , ) (Ku − f (Ku − f from which it follows that (18) is reduced to ⎫ ⎧ ⎬ ⎨ −KK Kf u u d ≥0 . U = u ∈ Rn f K α2 − f ⎭ ⎩ 1 f 1
(19)
Here, we have utilized K = K . The constraints in Problem (9) are written as u∈U
=⇒
6
u ∈ F.
(20)
By using (13), (19), S-Lemma (Lemma 2.3) and the homogenization (Lemma 2.2), we see that (20) holds if and only if −KK K f −dl d v d l l l (l = 1, . . . , nc ). (21) τl ∃τl ≥ 0 s.t. c )2 − v 2 2−f vl d (u f K α f l l l Notice here that dl d l O leads to
vl dl −dl d l c vl dl (ul )2 − vl2
O,
which implies that τl = 0 does not satisfy (21). Hence, by putting ρl =
1 τl
(l = 1, . . . , nc ),
the implication (21) is reduced to −ρl dl d ρl vl dl − K f l + KK O ∃ρl ≥ 0 s.t. f ρl [(ucl )2 − vl2 ] − α2 + f ρl vl d l −f K
(l = 1, . . . , nc ).
Consequently, Problem (9) is reduced to α (a, uc ) = max α : Gl (α2 , ρ, a) O,
ρl ≥ 0 (l = 1, . . . , nc ) .
(22)
We see in Problem (22) that maximizing α is equivalent to maximizing α2 , which concludes the proof. The reduction of (20) to (21) is motivated by an extension of the idea found in the proof of Theorem 1 in [20]. It should be emphasized that Problem (17) is embedded into the dual SDP problem (11), where n = nc + nc (nd + 1) and m = nc + 1. Hence, the size of Problem (17) is bounded by the polynomial of nd and nc , whereas the original problem (9) has the infinitely many constraints. Moreover, it is known that an SDP problem can be solved efficiently by using the primal-dual interiorpoint method [6], which finds the global optimal solution with the number of arithmetic operations bounded by a polynomial of n and m. 4. Maximization problem of robustness function. In Section 2.1, we observed that the truss with the larger robustness function can be regarded as to be more robust. We attempt in this section to find the vector of cross-sectional areas a which maximizes the robustness function α (a, uc ). We call this structural optimization problem the maximization problem of robustness function. The conventional constraint on structural volume is considered, namely, a should satisfy n m
i ai ≤ V .
(23)
i=1
Here, i > 0 and V > 0 denote the unstressed member length of the ith member and the specified upper bound of structural volume, respectively. 7
For the given uc and V , the maximization problem of robustness function associated with stress constraints is formulated as ⎫ ⎪ (a, uc ) maxm α ⎪ ⎬ ∈Rn m n (24) ⎪ i ai ≤ V . ⎪ s.t. a ≥ 0, ⎭ i=1
We may assume without loss of generality that there exists an a ∈ Rn satisfying α (a, uc ) > 0. Hence, the objective function of Problem (24) can be replaced with α (a, uc )2 , without changing the optimal solution. By regarding a as the parameters, consider the following parametric SDP problem: ⎫ ⎪ max c t ⎪ ⎪ (t,)∈R×Rn ⎪ ⎪ ⎬ c s.t. Gl (t, ρ, a) O, ρl ≥ 0 (l = 1, . . . , n ), (25) nm ⎪ ⎪ ⎪ ⎪ i ai ≤ V . a ≥ 0, ⎪ ⎭ m
i=1
It follows from Proposition 3.1 that α (a, uc )2 can be regarded as the optimal value function of the parametric SDP problem (25). Consequently, the maximization problem (24) of robustness function is equivalent to the maximization problem of optimal value function of the parametric SDP c m problem (25), which is explicitly formulated in variables (t, ρ, a) ∈ R × Rn × Rn as ⎫ ⎪ t maxc m ⎪ ⎪ (t,, )∈Rn +n +1 ⎪ ⎪ ⎬ c s.t. Gl (t, ρ, a) O, ρl ≥ 0 (l = 1, . . . , n ), (26) nm ⎪ ⎪ ⎪ ⎪ ai i ≤ V , a ≥ 0. ⎪ ⎭ i=1
The remainder of this section is devoted to presenting the algorithm which solves Problem (26). Note that Problem (26) has the nonconvex constraints such that the symmetric matrices defined as the nonconvex matrix functions of (t, ρ, a) should be positive semidefinite. To solve Problem (26), we next propose the sequential SDP method, which is an extension of the successive linearization method for standard nonlinear programming problems (see, e.g., [4]). For simplicity, define x ∈ c m c m c m Rn +n +1 and h : Rn +n +1 → Rn +n +1 by x = (t, ρ, a) ∈ R × Rn × Rn , ⎞ ⎛ ρ ⎟ ⎜ ⎟ ⎜ a ⎟. ⎜ h(x) = ⎜ nm ⎟ ⎠ ⎝ V − i ai c
m
(27)
i=1
In accordance with the definition (16) of Gl , we write Gl (x) = Gl (t, ρ, a)
(l = 1, . . . , nc ).
Then Problem (26) is simply rewritten as max x1 s.t. h(x) ≥ 0,
Gl (x) O 8
(l = 1, . . . , nc ).
(28)
Moreover, letting ∆x = (∆t, ∆ρ, ∆a), define F l : Rm → S n +1 (l = 1, . . . , nc ) by ∆ρ + K(∆a)K(a) + K(a)K(∆a) ∆ρ v f − K(∆a) f −dl d l l l l l . F l (∆x) = c ∆ρl [(u )2 − v 2 ] − ∆t ∆ρl vl f − f K(∆a) d
l
l
(29)
l
The following is the sequential SDP method solving Problem (28) based on the successive linearization method: Algorithm 4.1 (Sequential SDP method for Problem (28)). Step 0: Choose a0 ≥ 0 satisfying (23) and α (a0 , uc ) > 0, c0 > 0, cmax ≥ cmin > 0, and the tolerance > 0. Set k := 0. Step 1: Find an optimal solution (tk , ρk ) of Problem (17) with a = ak . Step 2: Find the (unique) optimal solution ∆xk := (∆tk , ∆ρk , ∆ak ) ∈ R × Rn × Rn of the subproblem ⎫ 1 ⎬ max ∆x1 − ck ∆x22 ∆ 2 (30) s.t. h(∆x + xk ) ≥ 0, F (∆x) + G (xk ) O (l = 1, . . . , nc ). ⎭ c
l
m
l
If ∆xk 2 ≤ , then STOP. Step 3: Set ak+1 := ak + ∆ak . Step 4: Choose ck+1 ∈ [cmin , cmax ]. Set k ← k + 1, and go to Step 1. Note that the sequence of robustness function { αk } := {(tk )1/2 } is generated by Algorithm 4.1. Essentially, Algorithm 4.1 is designed in the spirit of the successive linearization method for the so-called nonlinear semidefinite programming problems proposed by Kanzow et al. [5]. One of the authors proposed the sequential SDP method for topology optimization problems with the specified linear buckling load factor [22]. Remark 4.2. Let Rn+ ⊂ Rn and Ln+ ⊂ Rn denote the non-negative orthant and the second-order cone [6], respectively, which are defined as Rn+ = {p = (p1 , . . . , pn ) ∈ Rn |pi ≥ 0 (i = 1, . . . , n)}, Ln+ = {p = (p0 , p1 ) ∈ R1 × Rn−1 |p0 ≥ p1 2 }. In order to solve Problem (30) in Step 2 of Algorithm 4.1, we use the well-developed software based on the primal-dual interior-point method. Some of these softwares, e.g., SeDuMi [26] and SDPT3 [27], are designed to solve the primal-dual pair of SDP problems in the following forms: (P) : min c x (31) s.t. Ax = b, x ∈ K; (D) : max b y (32) s.t. c − A y ∈ K,
9
where nL
nL
nS
nS
K = Rn+ × L+1 × · · · × L+p × S+1 × · · · × S+q . R
In order to use SeDuMi [26], we have to reformulate Problem (30) in the form of Problem (32). By introducing the auxiliary variables (s1 , s2 ), we see that Problem (30) is equivalent to the following c m problem formulated in variables (∆t, ∆ρ, ∆a, s1 , s2 ) ∈ Rn +n +3 : 1 max ∆t − ck s2 2 s.t.
k
∆ρ + ρ ≥ 0,
n m
k
∆a + a ≥ 0,
−
nm i ∆ai + V − i ∆aki ≥ 0,
i=1
s1 ≥ (∆t, ∆ρ, ∆a), s2 s1 O, s1 1
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
i=1
0 −dl d O 0 l + ∆ρl ∆t (ucl )2 0 0 −1 m n K(ak )K i + K i K(ak ) −K i f ∆ai + K i 0 −f i=1 k k k −K(ak )f −ρl dl dl + K(a )K(a ) O + f K(ak ) ρk (uc )2 − tk + f −f l
l
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c ⎪ (l = 1, . . . , n ). ⎪ ⎭
(33)
which can be embedded into Problem (32) with K = Rn+ +n c
m +1
× Ln+ +n c
m +2
2 n × S+ × (S+
d +1
)n . c
5. Convergent analysis of Algorithm 4.1. In this section, we shall show that Algorithm 4.1 is globally convergent under certain assumptions. = 0. First, it is shown that Algorithm 4.1 is well-defined in Throughout this section, we assume f the sense that each step of the algorithm is feasible. We start with investigating the feasibility of Problem (17), which is solved in Step 1 of Algorithm 4.1. Proposition 5.1. (t, ρ) = 0 is a feasible solution of Problem (17) for any a ∈ Rn a ≥ 0. Proof. By using the definitions (16) of Gl , it suffices to show that K(a)K(a) −K(a)f O (l = 1, . . . , nc ). Gl (0, 0, a) = f f −f K(a)
m
satisfying
(34)
> 0 that (34) holds if and only if f It follows from Lemma 2.4 and f
f K(a) K(a)f O. S := K(a)K(a) − f f 10
(35)
For w ∈ Rn , define p ∈ Rn and q ∈ Rn by d
d
d
p = K(a)w,
q=
f . 2 f
(36)
From q2 = 1, we obtain p2 ≥ q p.
(37)
It follows from (36) and (37) that S defined in (35) satisfies w Sw = p p − (q p)2 ≥ 0 for any w ∈ Rn , which concludes the proof. d
We next show that Problem (30), which is solved in Step 2 of Algorithm 4.1, always has the unique optimal solution. Proposition 5.2. If (tk , ρk ) is a feasible solution of Problem (17) with a = ak satisfying ak ≥ 0 and (23), then Problem (30) has the unique optimal solution. Proof. Observe that (tk , ρk , ak ) is also a feasible solution of Problem (26), say, Problem (28). Hence, it is obvious that ∆x = 0 satisfies h(∆x + xk ) = h(xk ) ≥ 0, F l (∆x) + Gl (xk ) = Gl (xk ) O
(l = 1, . . . , nc ),
i.e., ∆x = 0 is a feasible solution of Problem (30). Consequently, we see that Problem (30) has the nonempty convex feasible set and the strongly convex objective function, which concludes the proof. It follows from Proposition 5.1 and Proposition 5.2 that Algorithm 4.1 is well-defined in the sense that all problems solved are feasible. In what follows, we make the following assumption: Assumption 5.3. Problem (30) is strictly feasible at each iteration. We next show that our termination criterion in Step 2 is satisfied if and only if the current iterate xk coincides with a stationary point of Problem (28). Lemma 5.4. If ∆xk = 0 is the (unique) solution of Problem (30) for some ck > 0, then xk is a stationary point of Problem (28). Conversely, if xk is a stationary point of Problem (28), then ∆xk = 0 is the unique solution of Problem (30) for any ck > 0. Proof. We first observe that the Karush–Kuhn–Tucker (KKT) conditions for Problem (28) can be written as nc −1 ∂Gl (x) ∂h(x) − µ = 0, Ul • − ∂x ∂x 0 l=1 (38) Gl (x) O, U l O, U l • Gl (x) = 0 (l = 1, . . . , nc ), h(x) ≥ 0,
µ ≥ 0,
µ h(x) = 0, 11
where (U 1 , . . . , U nc , µ) ∈ S n +1 × · · · × S n +1 × Rm are the Lagrange multipliers. Similarly, the KKT conditions of Problem (30) are written as nc k −1 ∂F l (∆x) ∂h(∆x + x ) k − µk = 0, − U kl • c ∆x + ∂∆x ∂∆x 0 l=1 (39) F l (∆x) + Gl (xk ) O, U kl O, U kl • (F l (∆x) + Gl (xk )) = 0 (l = 1, . . . , nc ), d
h(∆x + xk ) ≥ 0,
d
µk ≥ 0,
µk h(∆x + xk ) = 0,
where (U k1 , . . . , U knc , µk ) ∈ S n +1 × · · · × S n +1 × Rm are the Lagrange multipliers. Since Problem (30) is a convex programming problem with a strictly feasible set, ∆x is an optimal solution of Problem (30) if and only if there exists a set of U k1 , . . . , U knc , µk satisfying (39). From the definitions (16), (27), and (29) of Gl , h, and F l , we obtain d
d
∂h(x) ∂h(∆x + xk ) = , ∂∆x ∂x ∂Gl (x) ∂F l (∆x) = , F l (0) = O ∂∆x ∂x
(l = 1, . . . , nc ).
Consequently, if ∆xk = 0 is a solution of Problem (30), then the system (39) is reduced to (38), which concludes the proof. In what follows, we assume that Algorithm 4.1 generates an infinite sequence {xk } = {(tk , ρk , ak )}. Lemma 5.5. The sequence {tk } generated by Algorithm 4.1 is monotonically nondecreasing. Moreover, tk = tk+1 if and only if ∆xk = 0. Proof. Let (tk , ρk , ak ) be a given iterate and ∆xk := (∆tk , ∆ρk , ∆ak ) be the solution of the corresponding subproblem (30). We first show that ∆tk ≥
1 k c ∆xk 22 2
(40)
is satisfied. It follows from the proof of Proposition 5.2 that ∆x = 0 is feasible for Problem (30). Since ∆xk is a solution of Problem (30), we obtain 1 ∆tk − ck ∆xk 22 ≥ 0, 2 which implies (40). Next, we show that ∆xk = 0
=⇒
tk+1 > tk .
(41)
If ∆xk = 0, we see that (40) implies tk + ∆tk > tk . Therefore, we shall show that tk+1 ≥ tk + ∆tk , i.e., it suffices to show that (tk + ∆tk , ρk + ∆ρk ) is feasible for Problem (17) with a = ak + ∆ak . Since (∆tk , ∆ρk , ak ) is feasible for Problem (30), we see F l (∆xk ) + Gl (xk ) O,
(42)
h(xk + ∆xk ) ≥ 0.
(43)
12
Notice here that the constraint conditions of Problem (17) are written as k )K(∆ak ) 0 K(∆a O, ρkl + ∆ρkl ≥ 0 (l = 1, . . . , nc ). F l (∆xk ) + Gl (xk ) + 0 0
(44)
Observe that, from (5), we have K(∆ak )K(∆ak ) O for ∆ak ≥ 0. It follows from this fact and (42) and (43) that (44) is satisfied. As a consequence of (43) and (44), we see that (tk +∆tk , ρk +∆ρk ) is feasible for Problem (17) with K = K(ak + ∆ak ), which leads to (41). Finally, we shall show that ∆xk = 0
=⇒
tk+1 = tk .
Suppose that we modify Problem (30) by adding the constraints ∆a = 0. Since ∆xk = 0 is the unique optimal solution of the original problem (30), it is also the unique optimal solution of the modified problem. On the other hand, under the constraints ∆a = 0, we see F l (∆xk ) + Gl (xk ) = Gl (tk + ∆t, ρk + ∆ρ, ak ) (l = 1, . . . , nc ). Hence, the modified problem coincides with Problem (17), where a := ak = ak+1 . Consequently, (t, ρ) = (tk , ρk ) is an optimal solution of Problem (17) solved in Step 1 at the (k + 1)th iteration, i.e., tk+1 = tk . The following theorem states our main result in this section, which guarantees the global convergence property of Algorithm 4.1 under Assumption 5.3: Theorem 5.6. Let {xk } be a sequence generated by Algorithm 4.1. Then any accumulation point of {xk } is a stationary point of Problem (28). Proof. Let x∗ := (t∗ , ρ∗ , a∗ ) denote an accumulation point of {xk }. {xk }k∈K denotes a subsequence of {xk } converging to x∗ . We see from the proof of Lemma 5.5 that 1 tk+1 − tk ≥ ∆tk ≥ ck ∆xk 22 2
(45)
for all k ∈ K. Since {tk } is monotonically nondecreasing and bounded from above by, e.g., t∗ , we have tk+1 − tk → 0 (k → ∞) on the subsequence K. By using (45) and the fact that {ck }k∈K is bounded from below by cmin > 0, we obtain {∆xk }k∈K → 0. This also implies {ck ∆xk }k∈K → 0,
(46)
because {ck }k∈K is bounded from above by cmax . It follows from {xk }k∈K → x∗ , {∆xk }k∈K → 0, and by continuity that we have ∂Gl (x∗ ) ∂F l (∆xk ) → , ∂∆x ∂x
h(∆xk + xk ) → h(x∗ )
13
(k → ∞)
(47)
y (1)
(a)
∼f
(b)
(2)
x
0 (c)
Figure 1: 2-bar truss.
on K. From (46), we also have ck ∆xk → 0 (k → ∞)
(48)
on K. Recall that Problem (30) is an SDP problem (see Remark 4.2). Consider the dual (SDP) problem of Problem (30). It follows from the duality theorem of SDP (Ben-Tal and Nemirovski [6, Theorem 2.4.1]) that the dual problem has an optimal solution under Assumption 5.3. Moreover, (U k1 , . . . , U knc , µk ) is an optimal solution of the dual problem if and only if the KKT conditions (39) are satisfied. Hence, {U kl }k∈K (l = 1, . . . , nc ) and {µk }k∈K , i.e., the sequences of the Lagrange multipliers of the system (39), are bounded, and we may further assume without loss of generality that {U kl }k∈K → U ∗l (l = 1, . . . , nc ) and {µk }k∈K → µ∗ for some (U ∗1 , . . . , U ∗nc , µ∗ ) satisfying (39). Therefore, taking the limit k → ∞ on the subsequence K in the KKT conditions (39) of Problem (30), we obtain (38) from (47) and (48). Hence, we conclude that x∗ coincides with a stationary point of Problem (28).
6. Numerical experiments. The optimal designs with the maximal robustness functions are computed for various trusses by using Algorithm 4.1. In the following examples, the uncertainty set T is supposed to be defined by (14) and (15). In Steps 1 and 2 in Algorithm 4.1, the SDP problems are solved by using SeDuMi [26], which implements the primal-dual interior-point method for the linear programming problems over symmetric cones [6]. Computation has been carried out on Pentium M (1.5GHz with 1GB memory) with MATLAB Ver. 6.5.1 [28] and SeDuMi Ver. 1.05 [26]. 6.1. 2-bar truss. Consider a two-bar truss illustrated in Fig.1. The nodal coordinates at the initial unstressed state of nodes (a), (b), and (c) are specified as (x, y) = (1.0, 1.0), (0, 1.0), and (0, 0), respectively. are Nodes (b) and (c) are pin-supported; i.e., nd = 2 and nm = 2. The nominal external forces f
14
0.4 0.3 0.2
σ2
0.1 0 −0.1 −0.2 −0.3 −0.4
0
0.2
0.4
σ1
0.6
0.8
1
Figure 2: Stress states of the 2-bar truss with the initial design a = a0 for randomly generated ζ satisfying (14) and (15) with α = α (a0 , σ c ). 1 0.8 0.6 0.4
σ2
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −0.4
−0.2
0
0.2
0.4 σ1
0.6
0.8
1
Figure 3: Stress states of the 2-bar truss with the optimal design a = a∗ for randomly generated ζ satisfying (14) and (15) with α = α (a∗ , σ c ).
15
Table 1: Iteration history of optimization of the 2-bar truss by Algorithm 4.1. k
α = (tk )1/2
ak1
ak2
∆xk 2
0 1 2 3 4
7.0711 10.7981 15.0531 15.6888 15.6904
20.0000 25.2708 31.2883 32.1938 32.1895
40.0000 36.2730 32.0180 31.3777 31.3807
59.7803 92.4996 30.1372 1.6035 0.0167
= (10.0, 0). Consider the stress constraints of all members defined by (6), where σ c = 1.0 given by f i (i = 1, 2). The initial solution a0 for Algorithm 4.1 is given by a0 = (20.0, 40.0). In accordance with Proposition 3.1, we solve Problem (17) by using the primal-dual interior-point method to find α (a0 , σ c ) = 7.0711. The maximization problem (26) of robustness function is solved by using Algorithm 4.1. The upper bound of structural volume is given as V = 76.5685 so that the volume constraint (23) becomes active at a = a0 . We choose ck = 10−5 at each iteration of Algorithm 4.1, and set = 0.05. The iteration history is as listed in Table 1. The obtained optimal cross-sectional (a∗ , σ c ) = 15.6904. areas are a∗ = (32.1895, 31.3807), and the corresponding robustness function is α In order to verify this result, we randomly generate a number of ζ, and compute the corresponding member stresses σi . For the initial solution a0 , Fig.3 shows the obtained stress states (σ1 , σ2 ) with α(a0 , σ c )). It is observed from Fig.2 that a = a0 corresponding to the randomly generated ζ ∈ Z( the stress constraints (6) are satisfied for all possible ζ. The worst case of this example corresponds to ζ = (5.0, −5.0) , where the constraint σ1 ≤ σ1c becomes active. For the optimal solution a∗ , Fig.3 α(a∗ , σ c )). It is observed shows (σ1 , σ2 ) with a = a∗ computed from the randomly generated ζ ∈ Z( from Fig.3 that the stress constraints (6) are always satisfied, and the constraints σ1 ≤ σ1c , σ2 ≤ σ2c , σ2 ≥ −σ2c become active in the worst cases, i.e., the constraints on both members can happen to be active as the result of optimization. 6.2. 10-bar truss. Consider a truss illustrated in Fig.4. Nodes (a) and (b) are pin-supported at (x, y) = (0, 50.0) and (0, 0), respectively, where nd = 8 and nm = 10. The lengths of members in x- and y-directions, respectively, are 100.0 and 50.0. Suppose that the external force (1.0, 0) are applied at node (c) as . Consider the stress constraints of all members defined by (6), where the nominal external forces f σic = 1.0 (i = 1, . . . , nm ). For the initial solution of Algorithm 4.1, we assign a0 by a0i = 200.0 (i = 1, . . . , nm ). According to Proposition 3.1, we solve the SDP problem (17) by using the primal-dual interior-point method to obtain α (a0 , σ c ) = 0.2575. The maximization problem (26) of robustness function is solved by using Algorithm 4.1. We set = 1.0, ck = 10−5 (∀k), and V = 1.8944 × 105 so that (23) becomes active at a0 . The optimal design found by Algorithm 4.1 is shown in Fig.5, where the width of each member is proportional to its cross-sectional area. The corresponding robustness function is α (a∗ , σ c ) = 0.72383, where 77 iterations are spent by the algorithm. The mean and standard deviation of CPU time, respectively, of one increment are 0.82 sec and 0.21 sec. 16
y (a)
(3)
(4)
(7)
(8)
(1) (9)
0
(b)
(2) (10) (6)
(5)
(c)
∼f
x
Figure 4: 10-bar truss.
Figure 5: Optimal design of the 10-bar truss.
In order to verify these results, we randomly generate a number of ζ ∈ Z( α), and compute 0 the corresponding member stresses σi . Fig.6 shows σi for a = a computed from the randomly generated ζ ∈ Z( α(a0 , σ c )). It is observed from Fig.6 that, at a = a0 , the worst case corresponds to ζ such that σ5 ≤ σ5c becomes active. For the optimal design a∗ , Fig.7 shows σi corresponding to the randomly generated ζ ∈ Z( α(a∗ , σ c ). From Fig.7, we see that the constraint |σi | ≤ σic of each member happens to become active as the result of optimization. 6.3. 29-bar truss. Consider a truss illustrated in Fig.8. Nodes (a) and (b) are pin-supported at (x, y) = (0, 100.0) and (0, 0), respectively, where nd = 20 and nm = 29. The lengths of members both in x- and y-directions are 50.0. Suppose that (0, −1.0) are applied at nodes (c) and (d) as the nominal . Consider the stress constraints of all members defined by (6), where σ c = 5.0 external forces f i (i = 1, . . . , nm ). The maximization problem (26) of robustness function is solved by using Algorithm 4.1, where a0i = 20.0 (i = 1, . . . , nm ). By using Proposition 3.1, the robustness function at the initial solution (a0 , σ c ) = 7.2613 × 10−2 . In Algorithm 4.1, we set = 0.1, ck = 10−5 (∀k), and a = a0 is found as α V = 3.3971 × 104 so that (23) becomes active at a = a0 . The optimal design a∗ found by Algorithm 4.1 after 53 iterations is shown in Fig.9, and the corresponding robustness function is α (a∗ , σ c ) = 1.1071. The mean and standard deviation of CPU time of one increment are 7.80 sec and 1.21 sec, respectively. Fig.10 and Fig.11 show σi with α(a0 , σ c )) and a = a0 and a = a∗ , respectively, corresponding to the randomly generated ζ ∈ Z( ζ ∈ Z( α(a∗ , σ c )). It is observed from Fig.10 that the worst cases at a = a0 corresponds to ζ such that the constraints σ9 ≤ σ c and/or σ15 ≥ −σ c become active. On the other hand, from Fig.11, we can see that the stress constraints (6) of almost all members happen to become active at a = a∗ . Note that the actual worst-case behaviors cannot be exactly predicted, in general, by taking a rather 17
1 0.8 0.6 0.4
σi
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0
2
4
6 members (i)
8
10
Figure 6: Member stresses of the 10-bar truss with the initial design a = a0 for randomly generated ζ satisfying (14) and (15) with α = α (a0 , σ c ).
Figure 7: Member stresses of the 10-bar truss with the optimal design a = a∗ for randomly generated ζ satisfying (14) and (15) with α = α (a∗ , σ c ).
small number of random samples of ζ. Hence, we see in Fig.11 that the stress constraints (6) of some members with relatively small cross-sectional areas, namely, members (1), (2), (12), (13), and (14), do not happen to be active. 7. Conclusions. In this paper, based on the robustness function [1], we have proposed a novel concept as well as a global convergent algorithm for the robust structural optimization. The robustness function associated with stress constraints under uncertain external forces is defined as the optimal objective value of an optimization problem having finite number of variables and 18
y (a)
(10)
(9) (18)
(1)
(15)
0
(14)
(22) (4)
(23) (6)
(28)
(8)
(29) (17)
(16) (c)
(b)
(7)
(26)
(13)
(21) (27)
(5)
(25)
(12)
(2)
(20)
(19) (3)
(24)
(11)
(d)
∼f
x
∼f
Figure 8: 29-bar truss.
Figure 9: Optimal design of the 29-bar truss.
infinitely many constraint conditions. Particularly, we have investigated the uncertainty sets which are expressed via Euclidean norms of a vector of uncertainty parameters. By using the semidefinite programming relaxation technique, we have reformulated the present semi-infinite programming problem into the semidefinite programming problem. As a novel scheme of the robust truss optimization, we have introduced the maximization problem of robustness function associated with stress constraints, which has been formulated as the so-called nonlinear SDP problem, i.e., the maximization of a linear function under the constraints such that the symmetric matrix defined as the nonlinear function of the variables should be positive semidefinite. A sequential SDP approach has been presented, where the SDP problems are successively solved by the primal-dual interior-point method to obtain the optimal truss designs. The method has been shown to be globally convergent under certain assumptions. In the numerical examples of various trusses, we have illustrated that the optimal truss designs can be found without any difficulty by the proposed method. At the obtained optimal designs, it has been shown that the stress constraints of all members happen to become active as the results of
19
5 4 3 2
σi
1 0 −1 −2 −3 −4 −5 0
5
10
15 members (i)
20
25
30
Figure 10: Member stresses of the 29-bar truss with the initial design a = a0 for randomly generated ζ satisfying (14) and (15) with α = α (a0 , σ c ).
Figure 11: Member stresses of the 29-bar truss with the optimal design a = a∗ for randomly generated ζ satisfying (14) and (15) with α = α (a∗ , σ c ).
optimization. Besides the computational and theoretical efficiency, the proposed algorithm has the advantage such that, at each iteration of the algorithm, the SDP problems can be solved by using the well-developed softwares based on the primal-dual interior-point method. Therefore, our major task is limited to input the constant matrices and vectors defining the SDP problems, and no effort is required to develop any optimization software. References [1] Ben-Haim, Y., Information-gap Decision Theory, Academic Press, London, 2001. [2] Wolkowicz, H., Saigal, R., and Vandenberghe, L., Editors, Handbook of Semidefinite 20
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