A sequential convex programming algorithm for minimizing a sum of Euclidean norms with nonconvex constraints

Given p, q and a finite set of convex polygons hP1, . . . , PN i in R 3 , we propose an approximate algorithm to find an Euclidean shortest path starting at p then visiting the relative boundaries of the convex polygons in a given order, and ending at q. The problem can be rewritten as a variant of the problem of minimizing a sum of Euclidean norms: minp1,...,pN PN i=0 kpi − pi+1k, where p0 = p and pN+1 = q, subject to pi is on the relative boundary of Pi , for i = 1, . . . , N. The object function is convex but not everywhere differentiable and the constraint of the problem is not convex. By using a smooth inner approximation of Pi with parameter t, a relaxation form of the problem, is constructed such that its solution, denoted by pi(t), is inside Pi but outside the inner approximation. The relaxing problem is then solved iteratively using sequential convex programming. The obtained solution pi(t), however, is actually not on the relative boundary of Pi . Then a socalled refinement of pi(t) is finally required to determine a solution passing the relative boundary of Pi , for i = 1, . . . , N

A sequential convex programming algorithm for minimizing a sum of Euclidean norms with non-convex constraints Le Hong Trang ∗1,2, Attila Kozma1, Phan Thanh An2,3, and Moritz Diehl1,4 1 Electrical Engineering Department (ESAT-STADIUS) / OPTEC, KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven-Heverlee, Belgium 2 Center for Mathematics and its Applications (CEMAT), Instituto Superior T´ecnico, Universidade de Lisboa, A. Rovisco Pais, 1049-001 Lisboa, Portugal 3 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam 4 Institute of Microsystems Engineering (IMTEK), University of Freiburg, Georges-Koehler-Allee 102, 79110 Freiburg, Germany Abstract Given p, q and a finite set of convex polygons P1 , . . . , PN in R3 , we propose an approximate algorithm to find an Euclidean shortest path starting at p then visiting the relative boundaries of the convex polygons in a given order, and ending at q. The problem can be rewritten as a variant of the problem of minimizing a sum of Euclidean norms: minp1 ,...,pN N i=0 pi − pi+1 , where p0 = p and pN +1 = q, subject to pi is on the relative boundary of Pi , for i = 1, . . . , N . The object function is convex but not everywhere differentiable and the constraint of the problem is not convex. By using a smooth inner approximation of Pi with parameter t, a relaxation form of the problem, is constructed such that its solution, denoted by pi (t), is inside Pi but outside the inner approximation. The relaxing problem is then solved iteratively using sequential convex programming. The obtained solution pi (t), however, is actually not on the relative boundary of Pi . Then a so-called refinement of pi (t) is finally required to determine a solution passing the relative boundary of Pi , for i = 1, . . . , N . It is shown that the solution of the relaxing problem tends to its refined one as t → 0. The algorithm is implemented by Matlab using CVX package. Numerical tests indicate that solution obtained by the algorithm is very closed to a global one. Keywords— Sequential convex programming, shortest path, minimizing a sum of Euclidean norms, non-convex constrain, relaxation. 1 Introduction The problem of minimizing a sum of Euclidean norms arises in many applications, including facilities location problem [20] and VLSI (very-large-scale-integration) layout problem [1]. Many numerical algorithms for solving the problem were introduced (see [11, 13, 14, 20], etc). To solve ∗ email: le.hongtrang@esat.kuleuven.be 1 the unconstrained problems of minimizing a sum of Euclidean norms, Overton [11] gave an algorithm which has quadratic convergence under some given conditions. Based on polynomial-time interior-point methods, Xue and Ye [20] in 1997 introduced an efficient algorithm which computes an ǫ-approximation solution of the problem. Some applications in the Euclidean single facility location problems, the Euclidean multifacility location problems, and the shortest network under a given tree topology, were also presented. Qi et al. [13, 14] proposed two methods for solving the problem. A smoothing Newton method was introduced in 2000, the algorithm is globally and quadratically convergent. In 2002 by transforming the problem and its dual into a system of strongly semi-smooth equations, they presented a primal-dual algorithm for the problem by solving this system. For solving the problem of minimizing a sum of Euclidean norms with linear constraints, Andersen and Christiansen [2] proposed a Newton barrier method in which the linear constraints are handle by an exact penalty L1 . A globally and quadratically convergent method was recently introduced by Zhou [21] to solve this problem. All these problems are convex. In this paper we consider the problem of finding the shortest path starting a point p then visiting the relative boundaries of convex polygons in 3D, denoted by P1 , . . . , PN , in a given order and ending at q. The problem can be rewritten as a variant of problem of minimizing a sum of Euclidean norms: minp1 ,...,pN N i=0 pi − pi+1 , where p0 = p and pN +1 = q are fixed, pi is on the relative boundaries of the convex polygons, for i = 1, . . . , N. This problem is non-convex. Based on sequential convex programming approach [15], we introduce an approximate algorithm for solving the problem. By using a smooth inner approximation of Pi with parameter t, a relaxation form of the problem is constructed such that its solution, denoted by pi (t), is inside Pi but outside the inner approximation. The relaxing problem is then solved iteratively using sequential convex programming. The obtained solution pi (t), however, is actually not on the relative boundary of Pi . Then a so-called refinement of pi (t) is finally required to determine a solution passing the relative boundary of Pi , for i = 1, . . . , N. It is also shown that the solution of the relaxing problem tends to its refined one as t → 0. The algorithm is implemented by Matlab using CVX package. Numerical tests indicate that solution obtained by the algorithm is closed enough to global one. The rest of the paper is organized as follows. In section 2, we briefly recall the general framework of sequential convex programming, some notations, and a method for approximating convex polygons. Section 3 presents the formulations of the problem, and then introduces our new algorithm. Section 4 gives analysis of proposed algorithm. In section 5, some numerical tests are given. The conclusion is given in section 6. 2 Preliminaries 2.1 Sequential convex programming (SCP) We now recall a framework of SCP [15]. Consider the problem  min cT x    s.t. g(x) ≤ 0,    x ∈ Ω, 2 (1) where g : Rn → Rm is a nonlinear and smooth function on its domain, and Ω is a nonempty closed convex subset in Rn . The main challenge of the problem (1) is concentrated in the nonlinear of g(x). This can be overcome by linearizing it at current iteration point while maintaining the remaining convexity of the original problem. Let us assume that g(x) in twice continuously differentiable on its domain and denote by λ the Lagrange multiplier of Lagrange function of (1). The full-step sequential convex programming algorithm for solving (1) is given as follows. A general SCP framework 1. Choose an initial point x0 ∈ Ω and λ0 ∈ Rm . Let k = 0. 2. We solve iteratively the following convex problem  min cT x    s.t. g(xk ) + ∇g(xk )T (x − xk ) ≤ 0,    x∈Ω k k − z k ≤ ε holds for a given ε > 0, then := (xk+ , λk+ ). We check if z+ to obtain a solution z+ k stop. Otherwise, z k+1 := z+ and k := k + 1. 2.2 Notations We denote the Euclidean norm in Rn (n ≥ 2) with · . Let a ∈ R3 and A ⊂ R3 , the distance from a to A is given by a − a′ . d(a, A) = inf ′ a ∈A 3 Let A, B ⊂ R , the directed Hausdorff distance from A to B is defined by (see [7]) dH (A, B) = sup d(a, B) . a∈A Given a convex polygon P in R3 , we also recall the relative interior [18], denoted by riP , as follows riP = {p ∈ P : ∀q ∈ P, ∃λ > 1, λp + (1 − λ)q ∈ P }. Then the set ∂P := P \ riP, is said to be relative boundary of the polygon P . Let P1 , . . . , PN be a sequence of convex polygons in R3 and p1 , . . . , pN be a sequence of points where pi ∈ Pi , for i = 1, . . . , N. We define a refinement of the sequence p1 , . . . , pN into ∂Pi as follows. Definition 1. The sequence p¯1 , . . . , p¯N obtained by p¯i = arg min{ p′i − pi }, p′i ∈∂Pi is called the refined sequence of p1 , . . . , pN on ∂Pi . 3 for i = 1, . . . , N, (2) Given a convex polygon P in R3 , a parameter t > 0, and p ∈ P , a function denoted by Φ(p, t) is said to be an inner approximation of ∂P if the function satisfies the following: Φ(p, t) is a concave differentiable function ∀p ∈ P, the set PΦ(p,t) := {p|Φ(p, t) ≥ 0} ⊂ P, SPΦ(p,t) → SP as t → 0 monotonously, (3) where SPΦ(p,t) and SP denote the areas of closed regions bounded by PΦ(p,t) and P , respectively. Such approximation can be obtained by some methods, for example using KS-function [8] (described in next subsection), convex function approximation [6], and soft-max [4]. 2.3 Kreisselmeier-Steinhauser (KS) function The KS function was first introduced by Kreisselmeier and Steinhauser [8]. The function aims to present a single measure of all constraints in an optimization problem. In particular, consider an optimization problem containing m inequality constraints g(x) ≤ 0, the KS function overestimates a set of inequalities of the form y = gj (x), j = 1, . . . , N. A composite function is defined as fKS (x, ρ) = 1 ln ρ m eρgj (x) , j=1 where ρ is an approximate parameter. Raspanti et al. [17] have shown that for ρ > 0, fKS (x, ρ) ≥ max{gj (x)}. Furthermore, for ρ1 ≥ ρ2 , fKS (x, ρ1 ) ≤ fKS (x, ρ2 ). This implies that the fKS (x, ρ1 ) gives a better estimation of the feasible region of the optimization problem than fKS (x, ρ2 ). In the following example, the application of KS function for some single-variable functions is visualized. Consider two simple convex inequality constraints g1 (x, y) = (x − 5)2 − y ≤ 0, g2 (x, y) = x − y − 1 ≤ 0. The KS function of the constraints is shown in Fig. 1. Because the convexity of KS function is followed the original constraints, KS function is convex. Furthermore, fKS (x, ρ) tends to max{g1 (x), g2 (x)} as ρ → ∞ [17]. Intuitively, this means that fKS (x, ρ) approaches the boundary of the feasible region. 3 An SCP algorithm for finding the shortest path visiting the relative boundaries of convex polygons 3.1 Relaxing form Given p, q and a sequence of convex polygons P1 , P2 , . . . , PN in R3 , a formulation of the problem of finding shortest path visiting the relative boundaries of convex polygons, can be given by N min pi ∈∂Pi pi − pi+1 i=0 4 (PMSN ) 30 g1(x) = (x−5)2 25 g2(x) = x−1 t = 0.1 t = 0.2 t = 0.5 20 y 15 10 5 0 −5 0 2 4 6 8 10 x Figure 1: KS function for simple constraints. where p0 = p and pN +1 = q. There are two challenges for solving the problem. The first one is that pi is constrained to be on ∂Pi , then this constraint is not convex. Second, the function presenting the relative boundary is non-smooth at vertices of the polygons. Hence, the function is not differentiable at vertices of the convex polygons. These can be overcome by using a relaxation of the constrains of problem (PMSN ). Our method computes first an inner approximation of each ∂Pi using a function Φi (pi , t) satisfying (3). An intermediate solution denoted by pi (t), can be then obtained by using the sequential convex programming, described later, such that both following conditions are satisfied, Φi (pi , t) ≤ 0 and pi ∈ Pi , (4) for i = 1, 2, . . . , N. We then decrease the value of t to get better approximation of Pi , this aims to push pi (t) to ∂Pi . In Particular, pi (t) is generated such that pi (t) → p¯i (t) as t → 0, where p¯i (t) ∈ ∂Pi is obtained by using (2), for i = 1, 2, . . . , N (see Proposition 6 later). A geometrical interpretation of proposed method is shown in Fig. 2. Given two point p, q and a convex polygons P1 , P2 , we seek a point pi ∈ ∂Pi such that p − p1 + p1 − p2 + p2 − q is minimal, for i = 1, 2. Let us initialize t = t0 , an approximation Φi (pi , t0 ) of ∂Pi is computed. We then determine pi (t0 ) such that Φi (pi (t0 ), t0 ) ≥ 0 and pi (t0 ) ∈ Pi by using SCP. We reduce t = tj and compute iteratively pi (tj ), for j = 1, 2, . . ., until t is small enough. This means that pji will be moved closer to ∂Pi step by step. A sequence of intermediate solutions pi (tj ) (j ≥ 0) is determined such that Φi (pi (tj ), tj ) ≥ 0 and pi (tj ) ∈ Pi , for i = 1, 2. Instead of solving directly (PMSN ), we solve a sequence of sub-problems as follows  minp1 ,...,pN N  i=0 pi − pi+1   s.t. (PRLX (p, t))  Φi (pi , t) ≤ 0, i = 1, . . . , N,   pi ∈ Pi , i = 1, . . . , N, where p0 = p and pN +1 = q. We now formulate (PRLX (p, t)) for a certain value of t. With given 5 p p1(t0) P1 p1(t1) p1(t2) Φ2(p2, t2) ≤ 0 P2 Φ2(p2, t1) ≤ 0 p2(t0) p2(t1) p2(t2) q Figure 2: An example for illustrating the idea of our method. convex polygons Pi ⊂ R3 and pi ∈ Pi , for i = 1, . . . , N, this can be expressed by Ai pi + bi ≤ 0, cTi pi + di = 0, i = 1, . . . , N, i = 1, . . . , N, (5) where Ai and bi are respectively parameter matrix and vector. For each convex polygon Pi , a row of Ai together a corresponded element of bi and cTi pi + di = 0 specify a line equation containing a boundary edge of Pi . A numerical formulation of nonlinear optimization of problem (PRLX (p, t)) is thus given by  N  minp1 ,...,pN i=0 pi − pi+1     s.t. (PN LP (p, t)) Φi (pi , t) ≤ 0, i = 1, . . . , N,    A p + b ≤ 0, i = 1, . . . , N, i i i   cTi pi + di = 0, i = 1, . . . , N, where p0 = p and pN +1 = q. Remark 1. Let f (p1 , p2 , . . . , pN ) = N i=0 pi − pi+1 , where p0 = p and pN +1 = q, pi ∈ Pi for i = 1, . . . , N. Then f is a convex function. In problem (PN LP (p, t)), the first constraint is a concave. By using the SCP approach described in subsection 2.1, we take the linearization of Φi (pi , t) at p¯i by Φi (pi , t) ≃ Φi (p¯i , t) + ∇Φi (p¯i , t)T (pi − p¯i ). 6 (6) The solution of the problem (PN LP (p, t)) can then be obtained approximately by solving a sequence of the following subproblems  minp1 ,...,pN N  i=0 pi − pi+1     s.t. (PSCP (pk , t)) Φi (pki , t) + ∇Φi (pki , t)T (pi − pki ) ≤ 0, i = 1, . . . , N, k = 0, 1, . . . ,    Ai pi + bi ≤ 0, i = 1, . . . , N,   T ci p i + d i = 0, i = 1, . . . , N, where p0 = p, pN +1 = q, and p0i is an initial feasible value. 3.2 Approximating the convex polygons using the KS function We can approximate the convex polygons in problem (PN LP (p, t)) using KS function described in subsection 2.3. Let Ai := (Ai1 , . . . , Aimi )T , and bi := (bi1 , . . . , bimi )T where mi is the number of edges of Pi , for i = 1, . . . , N. In order to use KS function, we first set proper matrices Ai and bi then take mi 1 e t (Aij pi +bij ) , Φi (pi , t) := t ln (7) j=1 where t is the approximate parameter, for i = 1, . . . , N. The problem (PN LP (p, t)) is then rewritten as follows,  minp1 ,...,pN N  i=0 pi − pi+1     s.t. (PN LP−KS (p, t)) Φi (pi , t) ≤ 0, i = 1, . . . , N,    Ai pi + bi ≤ 0, i = 1, . . . , N,   cTi pi + di = 0, i = 1, . . . , N, where p0 = p and pN +1 = q. Taking linearization of Φi (pi , t) gives the following problem in the SCP approach.  minp1 ,...,pN N  i=0 pi − pi+1     s.t. Φi (pki , t) + ∇Φi (pki , t)T (pi − pki ) ≤ 0, i = 1, . . . , N, k = 0, 1, . . . , (PSCP−KS (pk , t))   Ai pi + bi ≤ 0, i = 1, . . . , N,    T = 0, i = 1, . . . , N, ci p i + d i where p0 = p, pN +1 = q, and p0i is an initial feasible value. 3.3 New algorithm Given p, q and a sequence of convex polygons P1 , P2 , . . . , PN in R3 , Algorithm 1 solves iteratively (PN LP−KS (p, t)) to obtain a refined local approximate shortest path passing the ∂Pi by means 7 of numerical solution of the corresponding nonlinear programming problem, for i = 1, 2, . . . , N. Namely, each step of the algorithm a local shortest path can be found with a certain value of t. The path, however, passes in the relative interiors of the convex polygons, then the refinement (2) is performed to ensure that it passes the relative boundaries of the polygons. By Definition 1 this is a refinement of the resulting shortest path. Proposition 6 indicates that the shortest path tends to its refinement as t → 0. Let us denote by p1 (t), . . . , pN (t) and p¯1 (t), . . . , p¯N (t) are respectively the SCP solution of (PN LP−KS (p, t)) and its refinement with a certain value of t. Algorithm 1 Finding the refined shortest path visiting the relative boundaries of convex polygons Input: Given p, q and P1 , P2 , . . . , PN in R3 , an error of solution ε > 0, step length 0 < η < 1, and a tolerance of SCP solver µ > 0. Output: π(p, q) = p, p¯1 (t), . . . , p¯N (t), q where p¯i (t) ∈ ∂Pi , for i = 1, . . . , N. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Initialize t ← t0 > 0, and p0i ∈ Pi , for i = 1, . . . , N. repeat Approximate ∂Pi by Φi (pi , t) using (7), i = 1, . . . , N. p(t) := (p1 (t), . . . , pN (t))T and Φ(p, t) := (Φ1 (pi , t), . . . , ΦN (pi , t))T . Call SCP S OLVER Φ(p, t), p to solve (PN LP−KS (p, t)) which gives p1 (t), . . . , pN (t) . Refine p1 (t), . . . , pN (t) to obtain p¯1 (t), . . . , p¯N (t) to be on ∂Pi using (2). p¯(t) := (¯ p1 (t), . . . , p¯N (t))T t ← ηt. until p(t) − p¯(t) ≤ ε. return π(p, q) := p, p¯1(t), . . . , p¯N (t), q . 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: procedure SCP S OLVER(Φ, p0 ) k ← 0. repeat Compute linearization of Φi (pi , t) at pki . Solve convex subproblem (PSCP−KS (pk , t)) to obtain pk+1 . i k ← k + 1. pk ← (pk1 , pk2 , . . . , pkN )T . until pk − pk−1 ≤ µ. return pk1 , pk2 , . . . , pkN . end procedure 4 Algorithm analysis Let p := (p1 , p2 , . . . , pN )T . We now analyze the algorithm. The following property shows the feasibility of full SCP step of the Algorithm 1. Proposition 1. For a certain value of t, if p′ (t) is a feasible point of the problem (PN LP−KS (p, t)) then a solution p∗ (t) of the problem (PSCP−KS (pk , t)) corresponding to p′ (t) exists and is feasible for the problem (PN LP−KS (p, t)). 8 Proof. Since p∗ (t) is a solution of (PSCP−KS (pk , t)), Ai p∗i (t) + bi ≤ 0 and cTi p∗i (t) + di = 0, for i = 1, . . . , N. We need only to show that Φi p∗i (t), t ≤ 0, for i = 1, . . . , N. Indeed, since Φi pi (t), t is concave, we have Φi p∗i (t), t ≤ Φi p′i (t), t + ∇Φi p′i (t), t T p∗i (t) − p′i (t) , for i = 1, . . . , N. Furthermore, p∗ (t) is a solution of (PSCP−KS (pk , t)), then Φi p′i (t), t + ∇Φi p′i (t), t T p∗i (t) − p′i (t) ≤ 0. It follows that Φi p∗i (t), t ≤ 0. The proof is completed. Proposition 2. Procedure SCP S OLVER gives a local solution of the problem (PN LP−KS (p, t)). Proof. In order to solve problem (PN LP−KS (p, t)), procedure SCP S OLVER first converts the problem using extra variables y1 , y2 , . . . , yN which gives the following equivalent problem,  minp1 ,...,pN ,y1 ,...,yN N  i=1 yi    s.t.    pi − pi+1 − yi ≤ 0, i = 0, . . . , N − 1, (8)  Φi (pi , t) ≤ 0, i = 1, . . . , N,     Ai pi + bi ≤ 0, i = 1, . . . , N,   cTi pi + di = 0, i = 1, . . . , N. this is a second-order cone program. By [15] the SCP framework in procedure SCP S OLVER gives a local solution of the non-linear problem (8). In vector form, let Φi (pi , t) cTi pi + di hi (pi , t) := and gi (pi ) := , maxj (Aij pi + bi ) cTi pi + di , for i = 1, . . . , N, j = 1, . . . , mi , where mi is the number of edges of Pi . Proposition 3. limt→0 hi (pi , t) − gi (pi ) = 0, for i = 1, . . . N. Proof. The following is proven in [17], lim Φi (pi , t) = t→0 max {Aij pi + bij }, for i = 1, . . . , N. j∈{1,...,mi } Hence, hi (pi , t) − gi (pi ) = |Φi (pi , t) − max {Aij pi + bij }|. j∈{1,...,mi } It follows that lim hi (pi , t) − gi (pi ) = 0, for i = 1, . . . N. t→0 The proof is completed. 9 This means that hi (pi , t) converges to gi (pi ) as t → 0, for i = 1, . . . , N. The following states that the achieved convergence is uniform. Proposition 4. hi (pi , t) converges uniformly to gi (pi ) as t → 0, for i = 1, . . . N. Proof. We have that Pi is compact. On the one hand, {Φ(pi , t)} is a sequence of continuous functions and converges to maxj∈{1,...,mi } {Aij pi + bi } on Pi . On other hand, by Property 3 in [17] we have Φ(pi , t1 ) ≥ Φ(pi , t2 ), ∀pi such that t1 ≥ t2 . By Theorem 7.13 in [19], we conclude that Φ(pi , t) converges continuously to maxj∈{1,...,mi } {Aij pi + bij }, for i = 1, . . . , N, i.e., for every ǫ > 0, there exists T > 0, such that for every pi ∈ Pi and t < T , we have |Φ(pi , t) − max {Aij pi + bij }| < ǫ. j∈{1,...,mi } Since hi (pi , t) − gi (pi ) = |Φi (pi , t) − max {Aij pi + bij }|, j∈{1,...,mi } hi (pi , t) − gi (pi ) < ǫ. It follows that hi (pi , t) converges uniformly to gi (pi ) as t → 0, for i = 1, . . . N. Combining this proposition and Theorem 7.9 in [19], we have following corollary. Corollary 1. ∀pi ∈ Pi , then lim sup { hi (pi , t) − gi (pi ) } = 0, t→0 pi ∈Pi Furthermore, set ui := pix , piy , Φi (pi , t) T and vi := pix , piy , maxj∈{1,...,mi } {Aij pi + bij } lim sup { ui − vi } = 0, t→0 pi ∈Pi T , then for i = 1, . . . , N. Under the uniform convergence of hi (pi , t) the following indicates that there is a corresponding convergence of their graphs with respect to directed Hausdorff distance. Proposition 5. limt→0 dH ∂PΦi (pi ,t) , ∂Pi = 0, for i = 1, . . . , N. Proof. For pi ∈ Pi , taking ui := pix , piy , Φi (pi , t) dH ∂PΦi (pi ,t) , ∂Pi = T ∈ ∂PΦi (pi ,t) , for i = 1, . . . , N, we have that sup d(ui , ∂Pi ) ui ∈∂PΦi (pi ,t) = sup ui ∈∂PΦi (pi ,t) Set vi := pix , piy , maxj∈{1,...,mi } {Aij pi + bij } T inf vi′ ∈∂Pi ui − vi′ . ∈ ∂Pi , for i = 1, . . . , N. By Corollary 1, lim sup { ui − vi } = 0. t→0 pi ∈Pi 10 (9) This means that for every ǫ > 0, there exists t0 > 0, such that for t < t0 , we have sup ui ∈∂PΦi (pi ,t) ,vi ∈∂Pi { ui − vi } < ǫ, for i = 1, . . . , N. (10) Combining (9) and (10) gives dH ∂PΦi (pi ,t) , ∂Pi ≤ sup ui ∈∂PΦi (pi ,t) ,vi ∈∂Pi { ui − vi } < ǫ, for i = 1, . . . , N. We conclude that lim dH ∂PΦi (pi ,t) , ∂Pi = 0, for i = 1, . . . , N. t→0 Step 6 of Algorithm 1 refines p1 (t), . . . , pN (t) to p¯1 (t), . . . , p¯N (t) which are on the relative boundaries of the convex polygons Pi by (2). The following property indicates that the SCP solution given by solving (PSCP−KS (pk , t)), tends to ∂Pi as t → 0, for i = 1, . . . , N. Hence, the resulted path is a refined shortest path. Proposition 6. Let us denote by pi (t) a SCP solution, for i = 1, . . . , N. We take p¯i (t) such that p¯i (t) = arg min{ p′i − pi (t) }, p′i ∈∂Pi (t) then lim pi (t) − p¯i (t) = 0, i = 1, . . . , N. t→0 Φi(pi, t′) ≤ 0 Φi (pi, t) ≤ 0 pi(t0) pi(t) p¯i(t) p∗i Figure 3: pi (t) tends p¯i (t) ∈ ∂Pi as t → 0. 11 Proof. Since pi (t) is a SCP solution, for i = 1, . . . , N, Φi (pi (t), t) ≤ 0. Furthermore Φi (pi , t) is decreasing continuous with respect to t > 0, for i = 1, . . . , N. Then there exists t′ > 0 and t′ ≤ t such that Φi pi (t), t′ = 0, i.e., pi (t) := pix (t), piy (t), Φi pi (t), t′ T ∈ ∂PΦi (pi ,t′ ) , (see Fig 3). On the one hand, by Proposition 5 lim dH ∂PΦi (pi ,t′ ) , ∂Pi = 0, t′ →0 for i = 1, . . . , N. On the other hand, pi (t) − p¯i (t) ≤ dH pi (t), ∂Pi ≤ dH ∂PΦ(pi ,t′ ) , ∂Pi , for i = 1, . . . , N. It follows that lim pi (t) − p¯i (t) = 0, t→0 i = 1, . . . , N. Theorem 1. Given p, q ∈ R3 , a sequence of convex polygons P1 , . . . , PN in R3 , Algorithm 1 gives a refined shortest path stating at p then visiting the relative boundary of Pi , for i = 1, . . . , N, and ending at q in √ 1 t0 p0 − p∗ ) (11) O N 3 N log2 ( ′ ) log 1 ( ) log 1 ( η µ µ ε α time, where µ′ is tolerance of the solution of convex subproblem. p0 and p∗ are initial point and local solution obtained by full step of procedure SCP Solver. α ∈ (0, 1) is a constant. Proof. By the analysis of the Primal-dual potential reduction algorithm√for solving a second-order cone programming problem given section 4.4 of [10], it takes O N 3 N log( µ1′ ) time to solve problem (PSCP−KS (pk , t)) in procedure SCP S OLVER. By Theorem 1 in [15] if we choose initial points p0 is closed enough to the solution p∗ of( PN LP−KS (p, t)) then the convergent rate of full step of SCP algorithm is linear. Let α be the linear contraction rate. Then the outer loop of the 0 ∗ procedure SCP S OLVER will take O log 1 ( p −p ) time. By the way, for an initial value t0 , the α µ 0 outer loop of Algorithm 1 takes O log 1 ( tε ) time. Hence, (11) is followed. η 5 Implementation 5.1 Numerical tests We test Algorithm 1 with the convex polygons which are generated randomly in the square having size of [−10, 10]. The initial point on each polygon is chosen randomly inside the polygons. The 12 algorithm was implemented in Matlab 7.13 (r2011b) and was executed in a laptop with Platform Ubuntu 10.04, Intel(R) Core(TM)2 Duo CPU P9400 2.4 GHz, 2GB RAM. To solve the convex subproblems, we use the CVX package 1 with Sedumi solver. We first show a simple test with only one convex polygon in which three values of approximate parameter of t are used for approximating the relative boundary of the polygon (see Fig. 4). We start the algorithm with a large value of t, say t1 , then decrease into smaller values t2 and t3 . The corresponding SCP solution p1 (ti ) thus approaches the relative boundary of the convex polygon, for i = 1, 2, 3. Convex polygon Shortest path p 10 8 p1(t2) 6 p1(t1) 4 2 p1(t3) 0 10 q 5 10 0 0 −5 −10 −10 Figure 4: The behavior of iterations of our algorithm corresponding to three values of t. Fig. 5 shows an example of our method which treats a problem including some parallel convex polygons. In this example we choose arbitrarily initial points. The figure shows the behavior of last iteration of our algorithm. The black circles on the figure show the track of full-step of SCP method. 1 Available at: http://cvxr.com/cvx 13 Convex polygons Approximate solution 30 25 20 15 10 5 10 0 10 5 5 0 0 −5 −5 −10 −10 Figure 5: An example testing with parallel convex polygons. For solving relaxed problem (PRLX (p, t)), we can also use a Matlab function called FMINCON. In order to compare with each other, the algorithm used in both SCP method and FMINCON is interior point method. Table 1 shows the test results of a larger set of polygons. The tolerance for solving the convex subproblems is µ = 10−5 . Run times shown in table indicate that our algorithm runs faster compared to FMINCON function. Table 1: Number of SCP iterations and run time comparing with FMINCON. FMINCON N Iterations 5 10 50 100 1000 56 237 78 93 157 SCP method Time (in sec.) 20.726550 159.450801 335.066874 1180.509008 130658.840711 14 Iterations Time (in sec.) 17 22 18 61 93 5.827492 11.83902 29.713664 189.829125 4053.933176 5.2 Global solution comparison In order to appreciate the solution obtained by our method, we compare the solution with global one. Let us recall the original problem N min pi ∈∂Pi i=0 pi − pi+1 , where p0 = p and pN +1 = q. Then global solution of the problem can be computed naively by solving a set of convex optimization problems including all problems in form of N min pi ∈ei i=0 pi − pi+1 , (PSU B ) where ei is an edge of ∂Pi , for i = 1, . . . , N. Global solution of the original problem is best one among all problems of (PSU B ). We use CVX package to solve the problem (PSU B ), then compare with the solution of the same problem obtained by our method. The results of practical computation show that solution obtained by our method is really closed to the global one. Fig. 6 shows an example in which we consider problem consisting of five triangles (i.e., N = 5). Moreover, the time Convex polygons Approximate solution Global solution 30 25 20 15 10 5 0 10 10 5 0 0 −5 −10 −10 Figure 6: The solution obtained by SCP method is closed to the global solution. of computing global solution should be too largeas the number of convex polygons is large. This is caused by a large number of problems (PSU B ). We test with several simple data sets in which convex polygons are triangles. Table 2 shows the values of objective functions and run time of methods as well. For each problem, the values of objective function obtained respectively by two methods are very closed to 15 each other. The run time of finding global solution, however, is huge comparing with our method as the number of triangles is increased. Table 2: Run time (in seconds) of SCP method comparing with finding global solution. Numbers N 1 2 3 4 5 6 7 8 9 10 # 6 Global solution SCP solution of (PSU B ) f (p1 , . . . , pN ) Time f (p1 , . . . , pN ) Time 3 32 33 34 35 36 37 38 39 310 11.2932852761 25.3764906159 21.4718456914 30.6756720025 38.5995405264 37.1193471840 51.9773250115 58.2935123031 67.3492330551 # 0.786699 3.058703 10.640263 36.430685 122.666906 328.787751 1142.205860 3564.068292 12043.575363 # 11.2932852782 25.3764906160 21.4718457135 30.6756717918 38.5995404673 37.1193471990 51.9773249957 58.2935122392 67.5274566902 59.8034756238 4.310667 3.912728 5.213360 4.208676 4.696589 7.444731 8.114511 10.891497 8.045229 12.412469 Run time exceeded. Concluding remarks This paper proposed an approximate algorithm for solving a variant of problem of minimizing a sum of Euclidean norms with non-convex constraints. The problem is to find an Euclidean shortest path between two points visiting the relative boundaries of convex polygons in R3 . Based on KS function we transformed the problem into a relaxed form. The algorithm solves alternatively a sequence of the relaxed problems using a numerical optimization strategy called sequential convex programming and then gives a refined solution. Our method was implemented in Matlab. The results of practical computation indicated that solution obtained by our method is really closed to the global one. Acknowledgement This research was supported by Research Council KUL: PFV/10/002 Optimization in Engineering Center OPTEC, GOA/10/09 MaNet and GOA/10/11 Global real- time optimal control of autonomous robots and mechatronic systems. Flemish Government: IOF/KP/SCORES4CHEM, FWO: PhD/postdoc grants and projects: G.0320.08 (convex MPC), G.0377.09 (Mechatronics MPC); IWT: PhD Grants, projects: SBO LeCoPro; Belgian Federal Science Policy Office: IUAP P7 (DYSCO, Dynamical systems, control and optimization, 2012-2017); EU: FP7-EMBOCON (ICT- 248940), FP7-SADCO ( MC ITN-264735), FP7-TEMPO, ERC ST HIGHWIND (259 166), Eurostars SMART, ACCM. The first author (the third author, respectively) thanks the financial supports offered by Portuguese National Funds through FCT (Fundac¸a˜ o para a Ciˆencia e a Tecnologia) under the scope 16 of project PEst-OE/MAT/UI0822/2011 (National Foundation for Science and Technology Development, Vietnam (NAFOSTED) under grant number 101.02-2011.45 and the Vietnam Institute for Advanced Study in Mathematics (VIASM), respectively). References [1] J. Alpert, T. F. Chan, A. B. Kahng, I. L. Markov, and P. Mulet, Faster minimization of linear wirelength for global placement, IEEE Transaction on Computer-Aided Design Integrated Circuits and Systems, 17, 1998, pp. 3-13. [2] K. D. Andersen and E. Christiansen, A Newton barrier method for minimizing a sum of Euclidean norms subject to linear equality constraints, Technical Report, University of Southern Denmark, 1995. [3] K. F. Bloss, L. T. Biegler, and W. E. Schiesser, Dynamic process optimization through adjoint formulations and constraint aggregation, Industrial & Engineering Chemistry Research, 38(2), 1999, pp. 421-432. [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. [5] J. Fliege and S. Nickel, An interior point method for multifacility location problems with forbidden regions, Studies in Location Analysis, 14, 2000, pp. 23-45. [6] J. J. Koliha, Approximation of Convex Functions, Real Analysis Exchange, 29(1), 2003/2004, pp. 465-471. [7] C. Knauer, M. Lffler, M. Scherfenberg, and T. Wolle, The directed Hausdorff distance between imprecise point sets, Theoretical Computer Science, 412, 2011, pp. 4173-4186. [8] G. Kreisselmeier and R. Steinhauser, Systematic control design by optimizing a vector performance index, in IFAC Symposium on Computer-Aided Design of Control Systems (Ed. M. A. Cuenod), Zurich, Switzerland, Pergamon Press, Oxford, 1979, pp. 113-117. [9] A. Kr¨oller, T. Baumgartner, S. P. Fekete, and C. Schmidt, Exact solutions and bounds for general art gallery problems, ACM Journal of Experimental Algorithmics, 17(1), Article 2.3, 2012. [10] M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra and Its Application, 284, 1998, pp. 193-228. [11] M. L. Overton, A quadratically convergent method for minimizing a sum of Euclidean norms, Mathematical Programming, 27, 1983, pp. 34-63. [12] V. Polishchuk and J. S. B. Mitchell, Touring convex bodies - a conic programming solution, in: Proceedings of the 17th Canadian Conference on Computational Geometry, 2005, pp. 290293. 17 [13] L. Qui and G. Zhou, A smoothing Newton method for minimizing a sum of Euclidean norms, SIAM Journal on Optimization, 11(2), 2000, pp. 389-410. [14] L. Qui, D. Sun, and G. Zhou, A primal-dual algorithm for minimizing a sum of Euclidean norms, Journal of Computational and Applied Mathematics, 138, 2002, pp. 127-150. [15] T. D. Quoc and M. Diehl, Local convergence of sequential convex programming for nonlinear programming, in: Diehl, M.; Glineur, F.; Jarlebring, E.; Michiels, W. (Eds.), Recent advances in optimization and its application in engineering, Springer-Verlag, 2010, pp. 93-102. [16] T. D. Quoc and M. Diehl, Sequential convex programming methods for solving nonlinear optimization problems with DC constraints, http://arxiv.org/abs/1107.5841v1, 2011, pp. 1-18. [17] C. Raspanti, J. Bandoni, and L. Biegler, New strategies for flexibility analysis and design under uncertainty, Computers and Chemical Engineering, 24, 2000, pp. 2193-2209. [18] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [19] W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976. [20] G. Xue and Y. Ye, An efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM Journal on Optimization, 7(4), 1997, pp. 1017-1036. [21] G. Zhou, A quadratically convergent method for minimizing a sum of Euclidean norms with linear constraints, Journal of Industrial and Management Optimization, 3(4), 2007, pp. 655670. 18 [...]... Concluding remarks This paper proposed an approximate algorithm for solving a variant of problem of minimizing a sum of Euclidean norms with non -convex constraints The problem is to find an Euclidean shortest path between two points visiting the relative boundaries of convex polygons in R3 Based on KS function we transformed the problem into a relaxed form The algorithm solves alternatively a sequence of the... package 1 with Sedumi solver We first show a simple test with only one convex polygon in which three values of approximate parameter of t are used for approximating the relative boundary of the polygon (see Fig 4) We start the algorithm with a large value of t, say t1 , then decrease into smaller values t2 and t3 The corresponding SCP solution p1 (ti ) thus approaches the relative boundary of the convex. .. 17th Canadian Conference on Computational Geometry, 2005, pp 290293 17 [13] L Qui and G Zhou, A smoothing Newton method for minimizing a sum of Euclidean norms, SIAM Journal on Optimization, 11(2), 2000, pp 389-410 [14] L Qui, D Sun, and G Zhou, A primal-dual algorithm for minimizing a sum of Euclidean norms, Journal of Computational and Applied Mathematics, 138, 2002, pp 127-150 [15] T D Quoc and M Diehl,... Vandenberghe, S Boyd and H Lebret, Applications of second-order cone programming, Linear Algebra and Its Application, 284, 1998, pp 193-228 [11] M L Overton, A quadratically convergent method for minimizing a sum of Euclidean norms, Mathematical Programming, 27, 1983, pp 34-63 [12] V Polishchuk and J S B Mitchell, Touring convex bodies - a conic programming solution, in: Proceedings of the 17th Canadian... 101.02-2011.45 and the Vietnam Institute for Advanced Study in Mathematics (VIASM), respectively) References [1] J Alpert, T F Chan, A B Kahng, I L Markov, and P Mulet, Faster minimization of linear wirelength for global placement, IEEE Transaction on Computer-Aided Design Integrated Circuits and Systems, 17, 1998, pp 3-13 [2] K D Andersen and E Christiansen, A Newton barrier method for minimizing a sum of Euclidean. .. Raspanti, J Bandoni, and L Biegler, New strategies for flexibility analysis and design under uncertainty, Computers and Chemical Engineering, 24, 2000, pp 2193-2209 [18] R T Rockafellar, Convex Analysis, Princeton University Press, 1970 [19] W Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976 [20] G Xue and Y Ye, An efficient algorithm for minimizing a sum of Euclidean norms. .. Diehl, Local convergence of sequential convex programming for nonlinear programming, in: Diehl, M.; Glineur, F.; Jarlebring, E.; Michiels, W (Eds.), Recent advances in optimization and its application in engineering, Springer-Verlag, 2010, pp 93-102 [16] T D Quoc and M Diehl, Sequential convex programming methods for solving nonlinear optimization problems with DC constraints, http://arxiv.org/abs/1107.5841v1,... our algorithm The black circles on the figure show the track of full-step of SCP method 1 Available at: http://cvxr.com/cvx 13 Convex polygons Approximate solution 30 25 20 15 10 5 10 0 10 5 5 0 0 −5 −5 −10 −10 Figure 5: An example testing with parallel convex polygons For solving relaxed problem (PRLX (p, t)), we can also use a Matlab function called FMINCON In order to compare with each other, the algorithm. .. sequence of the relaxed problems using a numerical optimization strategy called sequential convex programming and then gives a refined solution Our method was implemented in Matlab The results of practical computation indicated that solution obtained by our method is really closed to the global one Acknowledgement This research was supported by Research Council KUL: PFV/10/002 Optimization in Engineering... Kreisselmeier and R Steinhauser, Systematic control design by optimizing a vector performance index, in IFAC Symposium on Computer-Aided Design of Control Systems (Ed M A Cuenod), Zurich, Switzerland, Pergamon Press, Oxford, 1979, pp 113-117 [9] A Kr¨oller, T Baumgartner, S P Fekete, and C Schmidt, Exact solutions and bounds for general art gallery problems, ACM Journal of Experimental Algorithmics, 17(1), Article ... Concluding remarks This paper proposed an approximate algorithm for solving a variant of problem of minimizing a sum of Euclidean norms with non -convex constraints The problem is to find an Euclidean shortest... a sum of Euclidean norms, Journal of Computational and Applied Mathematics, 138, 2002, pp 127-150 [15] T D Quoc and M Diehl, Local convergence of sequential convex programming for nonlinear programming, ... Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976 [20] G Xue and Y Ye, An efficient algorithm for minimizing a sum of Euclidean norms with applications, SIAM Journal on Optimization,