2 ReSNA. Matlab. ReSNA. Matlab. Wikipedia [7] ReSNA. (Regularized Smoothing Newton. ReSNA [8] Karush Kuhn Tucker (KKT) 2 [9]

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1 c ReSNA ReSNA (Regularized Smoothing Newton Algorithm) ReSNA 2 2 ReSNA 2 ReSNA ReSNA 2 2 Matlab 1. ReSNA [1] 2 1 Matlab Matlab Wikipedia [7] ReSNA (Regularized Smoothing Newton Algorithm) ReSNA 10 [8] Matlab 2 [9] ReSNA ReSNA ReSNA Karush Kuhn Tucker (KKT) Nash [2, 3] Wordrop [4, 5] 2 [6] Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

2 ReSNA M- ReSNA ReSNA ReSNA SeDuMi SDPT3 SDPA [10 12] ReSNA 2 ReSNA s K s R s s 2 (Second-Order Cone: SOC) { {x R x 0} (s =1) K s := { (x1, x) R R } s 1 x 1 x 2 (s 2). 2 x y 2 x y 2 ReSNA 2 2 (Second-Order Cone Complementarity Problem: SOCCP) [8] Find (x, y, p) R n R n R l such that x K,y K,x y =0, (1) y = F (x, p), G(x, p) =0. F : R n R l R n G : R n R l R l K R n n i 2 K = K n 1 K n 2 K nm (2) n = n 1 + n n m n R n + := {x R n x 0} K 1 K 1 (Mixed Complementarity Problem: MCP) Find (x, y, p) R n R n R l such that x 0, y 0, x y =0, y = F (x, p), G(x, p) =0 SOCCP (1) K := K 1 K 1 (Nonlinear Complementarity Problem: NCP) Find (x, y) R n R n such that x 0, y 0, x y =0,y= F (x) (3) SOCCP (1) K := K 1 K 1 l =0 p G F SOCCP (1) (Linear Complementarity Problem: LCP) 2 (Nonlinear Second-Order Cone Program: NSOCP) θ : R t R g : R t R n h : R t R k minimize θ(z) subject to g(z) K,h(z) = 0 (4) Karush Kuhn Tucker (KKT) θ(z) g(z)x h(z)w =0 x K,g(z) K,x g(z) =0,h(z) =0 (5) 4 KKT (5) 4 g : R t R n g(z) :=( g 1(z),..., g n(z)) R t n Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

3 ( z ) p :=, l := t + k, F(x, p) :=g(z), ( w ) θ(z) g(z)x h(z)w G(x, p):= h(z) SOCCP (1) ReSNA ReSNA Matlab 3.1 ReSNA M- ReSNA.m SOCCP (1) [x,y,p] = ReSNA(FUNC,nabFUNC,K,el,x0,y0,p0) FUNC Γ:R n+l R n+l ( ) ( ) F (x, p) x Γ(z) =, z =. (6) G(x, p) p Γ M- Gm.m Gm.m ReSNA.m FUNC Gm.m n + l 1 el z x p Γ F G nabfunc Γ :R n+l R (n+l) (n+l) Γ ( ) ( ) xf (x, p) xg(x, p) x Γ(z) =, z = p F (x, p) p G(x, p) p Γ M- nabgm.m nabgm.m ReSNA.m nabfunc nabgm.m n + l 1 (n + l) (n + l) 1 Γ(z) nabfunc = [] Γ(z) K SOCCP (1) 2 K K 2 (2) K = K 3 K 1 K 2 K = [3,1,2] K = R 10 + K = ones(1,10) el l SOCCP (1) p G(x, p) 0 n + l p G(x, p) el = 0 x0 x (0) sum(k) x0 = [] [ 1, 1] n y0 y (0) sum(k) y0 = [] [ 1, 1] n p0 p (0) el p0 = [] [ 1, 1] l x0,y0,p0 el x,y,p SOCCP (1) (x, y, p) 3.2 ReSNA.m PROGRESS Matlab (PROGRESS= Y ) (PROGRESS= N ) Y tole 0 1 Step 1 H NR (w (k) ) tole tole diff Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

4 10 8. eta, eta bar, rho, sigma, kappa, kappa bar, kappa hat 1 η η ρ σ κ κ ˆκ ReSNA.m 3.3 ReSNA 2 MCP NCP LCP NSOCP SOCCP (1) ReSNA.m ReSNA.m ReSNA MLSOCCP ReSNA.m 2 Find (x, y, p) R n R n R l such that x K, y K, x y =0, y = M 11x + M 12p + q 1, M 21x + M 22p + q 2 =0 M 11 R n n M 12 R n l M 21 R l n M 22 R l l q 1 R n q 2 R l MLSOCCP ReSNA.m M,q,K,el,x0,y0,p0 FUNC nabfunc M q ( ) ( ) M11 M 12 q 1 M =, q = (7) M 21 M 22 q 2 NCP ReSNA.m (NCP) (3) NCP ReSNA.m FUNC,nabFUNC,n,x0,y0 FUNC,nabFUNC F : R n R n F : R n R n n n x y F (x) n NSOCP ReSNA.m 2 (NSOCP) (4) ReSNA me first! manual plugin.pdf 3.4 ReSNA 5 (nabla checker.m) ReSNA ReSNA FF.m Γ:R 5 R 5 2Az Γ(z) := + b +0.01z, (8) 1+exp( z Az) A = , [ b = ]. Γ(z) = 2A 1+exp( z Az) + (2Az)exp( z Az)(2Az) +0.01I (1 + exp( z Az)) 2 FFd.m 1 nabla checker.m 5,5 FF.m Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

5 1 nabla checker.m 2 (8) Γ SOCCP (1) ReSNA.m n =4 l =1 K = K 3 R + F : R 4 R R 4 G : R 4 R R (6) 1],1 Γ M- FF.m Γ M- FFd.m K = K 3 R + p l = = FF.m = 5 (Iteration Information) k,j,m 1 k j m mu,epsi,beta μ k ε k β k Hme, Hnr H μk,ε k (w (k) ) H NR (w (k) ) MLCP ReSNA.m MCP: Find (x, y, p) R 3 R 3 R 2 such that x 0, y 0, x y =0, y = M 11x + M 12p + q 1, M 21x + M 22p + q 2 =0 2 ReSNA.m M 11 = ,M12 = , [ ] [ ] M 21 =,M 22 =, [ [ ] q 1 = ],q2 = MLCP ReSNA.m M,q,2 (7) M R 5 5 q R 5 p Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

6 ϕ NR (x 1,y 1 ) Φ NR (x, y) :=., ϕ NR (x m,y m ) ϕ NR (x i,y i ):=x i P 0(x i y i ), P 0(z) :=max{0,λ 1}u {1} +max{0,λ 2}u {2}. λ 1 λ 2 z u {1} u {2} z 6 P 0(z) z 2 K n i Φ NR (x, y) =0 x K,y K,x T y =0 [14, 15] H NR : R n R n R l R 2n+l Φ NR (x, y) H NR (x, y, p) := F (x, p) y (9) G(x, p) SOCCP (1) H NR (x, y, p) = 0 (10) SOCCP (1) 3 MLCP ReSNA.m 4. ReSNA 4.1 SOCCP (1) x, y 2 (2) x =(x 1,...,x m ) R n 1 R nm y =(y 1,...,y m ) R n 1 R nm Φ NR : R n R n R n (10) Φ NR F G H NR (x, y, p) 4.2 Φ NR μ>0 Φ μ ϕ μ(x 1,y 1 ) Φ μ(x, y) :=., ϕ μ(x m,y m ) ϕ μ(x i,y i ):=x i P μ(x i y i ), P μ(z) :=γ μ(λ 1)u {1} + γ μ(λ 2)u {2}. 6 [6, 13, 14] Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

7 λ 1 λ 2 z u {1} u {2} z γ μ 3 (i) lim α ĝ(α) =0 (ii) lim α (ĝ(α) α) =0 (iii) 0 < ĝ (α) < 1 ĝ : R R γ μ(λ) :=μĝ(λ/μ) (11) ReSNA (i) (iii) ĝ(α) := 1 2 ( α2 +4+α ) (12) λ R lim μ 0 γ μ(λ) =max{0,λ} Φ μ 2 μ>0 Φ μ (x, y) R n R n lim μ 0 Φ μ(x, y) =Φ NR (x, y). Φ NR Φ μ μ F G ε>0 F ε(x, p) :=F (x, p)+εx, G ε(x, p) :=G(x, p)+εp, F ε : R n R l R n G ε : R n R l R l F ε G ε F G Γ= ( F G) 7 ( Fε Gε) ε> (9) H NR H μ,ε : R n R n R l R 2n+l 7 (z,z ) R n+l R n+l ( Γ(z) Γ(z ) ) (z z ) 0 8 ε > 0 (z, z ) R n+l R n+l ( Γ(z) Γ(z ) ) (z z ) ε z z 2 Φ μ(x, y) H μ,ε(x, y, p) := F ε(x, p) y G ε(x, p) (13) (μ, ε) (0, 0) H μ,ε(x, y, p) =0 μ ε 0 5. ReSNA [8] 5.1 (a) λ : R n [0, + ) min λ i I(z) i(z) (I(z) ) λ(z) := 0 (I(z) = ), λ i(z) (i =1, 2) z I(z) {1, 2} I(z) :={i λ i(z) 0} (b) μ : R n R n [0, + ] + (δ 1/2 or α =0) μ(α, δ) := 1 2 α δ (δ <1/2 and α 0). (c) γ + 0 : R R 1 (α>0) γ + 0 (α):= ĝ (0) (α =0) 0 (α<0). (11) γ μ α lim μ 0 γ μ(α) =γ + 0 (α) μ γ μ(α) γ + 0 (α) Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.

8 5.1. α R δ >0 ĝ (12) μ (0, μ(α, δ)) γ μ(α) γ + 0 (α) <δ. (14) 5.2 x (k) w (k) := y (k) p (k) 1. Step 0 η, ρ (0, 1) η (0,η] σ (0, 1/2) κ >0 ˆκ >0 w (0) R 2n+l β 0 (0, ) μ 0 := H NR (w (0) ) ε 0 := H NR (w (0) ) k := 0 Step 1 H NR (w (k) ) =0 Step 2 Step 2 Step 2.0 v (0) := w (k) R 2n+l j := 0 Step 2.1 ˆd (j) R 2n+l H μk,ε k (v (j) )+ H μk,ε k (v (j) ) T d =0. Step 2.2 H μk,ε k (v (j) + ˆd (j) ) β k w (k+1) := v (j) + ˆd (j) Step 3 Step 2.3 Step 2.3 m H μk,ε k (v (j) + ρ m ˆd(j) ) 2 (1 2σρ m ) Hμk,ε k (v (j) ) 2. m j := m τ j := ρ m j v (j+1) := v (j) + τ ˆd(j) j Step 2.4 Step 3 H μk,ε k (v (j+1) ) β k (15) w (k+1) := v (j+1) Step 3 j := j +1 Step 2.1 { μ k+1 := min κ H NR (w (k+1) ) 2,μ 0η k+1, μ ( λ(x (k+1) y (k+1) ), ˆκ H NR (w (k+1) ) )}, ε k+1 := min { κ H NR (w (k+1) ) 2,ε 0η k+1}, β k+1 := β 0η k+1. k := k +1Step 1 Step H μk,ε k (w (k+1) ) β k w (k+1) k j (15) v (j+1) = w (k+1) Step 3 μ λ Step 3 0 < η η<1 {β k} {μ k} {ε k} 0 μ 0η k+1 ε 0η k+1 β 0η k+1 κ H NR (w (k+1) ) 2, μ ( λ(x (k+1) y (k+1) ), ˆκ H NR (w (k+1) ) ), κ H NR (w (k+1) ) 2 2 [8] SOCCP (1) l =0 p G 5.1. SOCCP (1) l =0 p G F { H μk,ε k (w (k) )} 1 {w k } SOCCP (1) 2 OR [1] [2] Hayashi, S., Yamashita, N. and Fukushima, M., Robust Nash equilibria and second-order cone complementarity problems, Journal of Nonlinear and Convex Analysis, 6, , [3] Nishimura, R., Hayashi, S. and Fukushima, M., Robust Nash equilibria in N-person noncooperative games: Uniqueness and Reformulation, Pacific Journal of Optimization, 5, , Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited

9 [4] Ito, Y., Robust Wardrop equilibria in the traffic assignment problem with uncertain data, Master s thesis, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, [5] Ordóñez, F. and Stier-Moses, N. E., Robust Wardrop equilibrium, Network Control and Optimization, Lecture Notes in Computer Science, 4465, , [6] 59, , [7] [8] Hayashi, S., Yamashita, N. and Fukushima, M., A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM Journal on Optimization, 15, , [9] Kanno, Y., Martins, J. A. C. and Pinto da Costa, A., Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem, International Journal for Numerical Methods in Engineering, 65, 62 83, [10] Sturm, J. F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11/12, , [11] Toh, K. C., Tütüncü, R. H. and Todd, M. J., SDPT3 a matlab software package for semidefinite programming, Optimization Methods and Software, 11, , [12] Yamashita, M., Fujisawa, K., Fukuda, M., Kobayashi, K., Nakata, K. and Nakata, M., Latest developments in the SDPA family for solving large-scale SDPs, Handbook on Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, Anjos, M. F. and Lasserre, J. B. (eds.), Springer, pp , [13] Faraut, J. andkorányi, A., Analysis on Symmetric Cones, Clarendon Press, [14] Fukushima, M., Luo, Z.-Q. and Tseng, P., Smoothing functions for second-order cone complementarity problems, SIAM Journal on Optimization, 12, , [15] Copyright c by ORSJ. Unauthorized reproduction of this article is prohibited.