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Applied Mathematical Sciences, Vol. 5, 2011, no. 2, 91 - 101 Multi-Criteria Linear-Quadratic Control Problem by KSH-Direction Interior-Point Method Kanuengkid, W. and Mouktonglang, T.1 Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai, Thailand Abstract We describe an implementation of an infinite dimension interiorpoint method for solving multi-criteria linear -quadratic control problem based on KSH-direction. It is shown that such a descent direction is independent on the scaling point. Each iteration requires solving finite rank perturbation of linear-quadratic control problems and a system of algebraic equations. The numerical result shows rapid convergence to the optimal solution. Mathematics Subject Classification: 47H09; 47H10 Keywords: Infinite dimension interior point method; Multi-criteria linearquadratic control; KSH-direction 1 Introduction In recent years, there are a lot of developments on generalizations of interiorpoint algorithms for linear programming (LP). The theories have been extended to the context of semidefinite programming (SDP) and more on several second-order cone problem. Recently, this class of problems gets more attention because of its wide applications. Now we study the infinite-dimensional optimization problem and a primal-dual algorithm based on Kojima-ShindolHara direction (KSH direction). We implement the algorithm for solving multi - criteria linear-quadratic control problem, that is a min-max optimization problem with linear constraints in the Hilbert space. This problem was recently implemented by using NT-Direction. For more detail, see [1]. It is well known that the NT-Direction is a member of both the KSH and MZ family. See [4] for more details. However, the NT-direction requires the, so called, scaling point on each iteration. The KSH-direction which was developed by 1 Corresponding author: [email protected] 92 Kanuengkid, W. and Mouktonglang, T. Kojima, Shindol and Hare is, on the other hand, independent on a scaling point. The existence and uniqueness of the direction is shown in [3]. This paper is organized as follows. In section 2, we introduce the general primal - dual optimization and Jordan algebras of finite rank for the analysis of an infinite-dimensional situations and describe the multi- criteria linear-quadratic control problem. In section 3, We state and prove the technical results by using KSH- direction. Finally, numerical results and conclusion are given in section 4. 2 Primal-dual interior-point algorithms In this section following [1], we describe a general set up for a class of infinite-dimension optimization problems. We then proceed with the optimality criterion and describe a primal-dual algorithm based on KSH direction. Let H be the Hilbert space. Consider the following optimization problem: t −→ min Wi (y − yi )H ≤ t i = 1, . . . , m, y ∈ c + Z. (1) Here .H is the norm induced by the scalar product |H .yi ∈ H are fixed. Z is a closed vector subspace in H, and Wi : H → H is a bounded linear operator for i = 1, . . . , m. The problem (1) is an example of infinite-dimensional second-order cone programming problem which has been analyzed in detail in [1]. The problem(1) can be easily rewritten in the conic form by following the substitution. Let V1 be a vector space. V1 = R × H. Let, further, V = V1 × · · · × V1 (m-times), X ⊂ V be a closed vector subspace in V . Ω1 = {(s, y) ∈ R × H : s ≥ yH } is the second-order cone which is known to be an open convex set in V1 . Ω = Ω1 × Ω1 × · · · × Ω1 (m − times). Let, further, Λ : V1 → V be a linear operator such that Λ(s, y) = {(s, W1 y), . . . , (s, Wm y)}. Then we define constants elements a and b in the vector space V and a closed vector space X ∈ V . a = ((1, 0), (0, 0), . . . , (0, 0)) ∈ V, b = ((0, W1 (c − y1 )), . . . , (0, Wm(c − ym ))), and X = Λ(R × Z). Naturally, the scalar product <, >V in V is defined as follows: < (t1 , x1 ), . . . , (tm , xm ), (s1 , y1), . . . , (sm , ym ) >V = m (si ti + xi , yiH ). i=1 With these notations, we can rewrite the problem (1) in the general optimiza- 93 KSH LQ-control problem tion problem < a, z >V → min, (2) z ∈ (b + X) ∩ Ω, and its dual < b, w >V → min, (3) ∗ w ∈ (a + X ⊥ ) ∩ Ω , Proposition 2.1. We have m Wi∗ ui ∈ Z ⊥ X = (r1 , u1), . . . , (rm , um ) ∈ V ; r1 + . . . , +rm = 0, ⊥ i=1 (4) where Z ⊥ is the orthogonal complement of Z and Wi∗ is the adjoint of Wi for each i. Consequently, a conic dual to (2) will have the following form [1]: m m < Wi (c − yi), ui >H → min (5) i=1 Wi∗ ui ∈ Z ⊥ , ui ≤ ri i = 1, . . . , m, i=1 m ri = 1. i=1 Here Wi∗ is the adjoint of Wi . The following theorem is a particular case of the result in [1] ∗ Theorem 2.2. Let int(F) = [(x + X) ∩ int(Ω)] × [(a + X ⊥ ) ∩ int(Ω )] = 0. Then both equatiom (2) have optimal solutions. Moreover, z ∗ is an optimal solution to equation (3) andw ∗ is an optimal solution to equation (2) and (3) if and only if < z ∗ , w ∗ >V = 0 One of the most important steps in implementation of primal-dual algorithm is a computation of a descent direction which drives duality gap μ to zero. One of these direction (The so-called KSH direction) is described later. 94 2.1 Kanuengkid, W. and Mouktonglang, T. Some Jordan-algebraic properties of spin-factors In what follows we restrict ourselves to the analysis of problem (2) and (3) for the case where V is a JB-algebra of a finite rank. Then we describe some Jordan-algebraic aspects of a spin-factor R × H essential for future considerations. (See [1] for more detail) Let V be the real commutative algebra with the unit element e. Given z ∈ V, consider the multiplication operator L(z) : V → V , L(z)z1 = z ◦ z1 , z1 ∈ V, Definition 2.3. We say that V is a Jordan algebra if the identity [L(z), L(z 2 )] = L(z)L(z 2 ) − L(z 2 )L(z) = 0 ∈ V, hold for anyz ∈ V. We can introduce the so-call quadratic representation in an arbitrary Jordan algebraV Given z ∈ V, P (z) = 2L(z)2 − L(z 2 ) = 0 ∈ V, s Let z = (s, y) ∈ R × H we write (s, y) ∈ V as a column vector . Then y each linear operator on R × H admits the following block partition: α A , B C where α ∈ R, A : H → R, B : R → H, A : H → H. Then α A s αs + Ay . = B C y Bs + Cy Given y ∈ H, introduce notation: ly : H → R, ly (y1 ) = (y|y1), y1 ∈ H. Observe that lyT : R → H, has the form: lyT (s) = sy, s ∈ R Here lyT is the transpose of ly with respect to the given scalar product (.|.) on Y and the standard scalar product on R i.e., sly (y1 ) = (lyT (s)|y1), s ∈ R, y ∈ H. With this notation, we have the following properties. 95 KSH LQ-control problem Proposition 2.4. Let z = (s, y) ∈ R × H. Then a multiplication linear operator L(z) is defined follow: s ly , L(z) = T ly sIH Here IH is the identity operator on H. Proposition 2.5. Let z = (s, y) ∈ R × H, y = 0. Consider y y 1 1 1, , e2 = 1, − 2 y 2 y λ1 = s + y, λ2 = s − y, y = (y|y). e1 = Then (s, y) = λ1 e1 + λ2 e2 is a spectral decomposition of z. And we define a determinant of z as follows: det(z) = s2 − y2. Then if det(z) = 0 the inverse z −1 of z is given by the formula z −1 = 1 (s, −y) = λ1 (z)−1 e1 + λ2 (z)−1 e2 , det(z) Next, we consider a function f (z) = −lndet(z) = +ln(s2 − y2 ). We next describe the quadratic representation in a spin-factorR × H. Given y ∈ H, we introduce a linear operator y ⊗ y as follows: y ⊗ y(y1) = (y|y1)y, y1 ∈ H Proposition 2.6. Let z = (s, y) ∈ R × H. Then a quadratic representation P (s, y) is defined follow: (y|y) sly , P (s, y) = det(z)IV + 2 slyT y ⊗ y Here IV is the identity operator on V = R × H For proof of the previous proposition see [1]. 96 Kanuengkid, W. and Mouktonglang, T. 2.2 Multi-criteria linear-quadratic control problem Denote by Ln2 [0, T ] the Hilbert space of square integrable function f : [0, T ] → Rn , T > 0. Let (x, u) ∈ H = Ln2 [0, T ] × Li2 [0, T ]. Consider the following optimization problem: T ((x − xi )T Qi (x − xi ) + (u − ui )T Ri (u − ui ))dt → min, J(x, u) = max i∈[1,m] 0 (6) ẋ(t) = Ax(t) + Bu(t), x(0) = x0 . where (xi , ui) ∈ H, i = 1, . . . , m, A, B, vQi , Ri , i = 1, . . . , m are given matrices and Qi , Ri are symmetric positive definite matrices. We can rewrite the problem (6) as a second order cone programming problem in(1). Here .H is the norm induced by the scalar product < | >H , yi = (xi , yi), i = 1, . . . , m; Z is the closed vector subspace in H described follows: Z = {(x, u); ẋ(t) = Ax(t) + Bu(t), T LQi 0 Wi = , i = 1, . . . , m, 0 LTRi x(0) = x0 } (7) where LTQi , LTRi are the lower-triangular matrices obtained by Cholesky factorization of Qi and Ri respectively. For more detail see [1,2] 3 KSH- direction The implementation of an infinite dimension interior point method for solving multi-criteria linear quadratic control problem described in the previous section can also be applied to other well known descent directions for instance KSH or KSH-dual [4]. The computation of a searching direction can be done in a similar way which requires solving finite rank perturbation of linear-quadratic control problem. This section devotes to the implementation of infinite dimension interior point method for solving multi-criteria linear quadratic control problem via KSH-direction. Given (n, d) ∈ Ω × Ω a pair of feasible solutions to the problem (2) and (3). Let (ξ, η) ∈ X × X ⊥ be KSH-direction. Hence (ξ, η) satisfies the following equation [4]. 1 1 1 1 1 η + P (n) 2 L(d)−1 P (n) 2 ξ = γP (n) 2 L(d)−1 P (n) 2 n−1 − P (n) 2 L(d)−1 d 1 = γP (n) 2 d−1 − n 1 where d = P (n)− 2 n , L(.) is a multiplication operator and P (.) is a quadratic representation. For simplicity let 1 Δ = γP (n) 2 d−1 − n. 97 KSH LQ-control problem Then the above equation is equivalent to 1 1 P (n)) 2 L(d)−1 P (n) 2 ξ − Δ ∈ X ⊥ which also equivalent to the following optimization problem: 1 1 < P (n) 2 L(d)−1 P (n) 2 ξ, ξ > − < Δ, ξ >→ min 2 ξ∈X (8) 1 We rewrite the equation (8) by using description of L(.)−1 and P (.) 2 suggested 1 in [1]. Note that each L(.)−1 and P (.) 2 can be written as a linear combina(1) (1) (m) (m) tion of rank one operators. Let Δ = (Δ0 , Δ1 ), . . . , (Δ0 , Δ1 ), then by combining all above terms, we obtain the following: 1 1 < P (s) 2 L(w)−1 P (s) 2 ξ, ξ > − < Δ, ξ > ρ(μ, ζ) = 2 2 m m m 1 γ (i) (i) 2 (i) (i) 2 = Wi ζ + β < a , Wi ζ > + ω (i) < u1 , Wi ζ >2 (i) 2 i=1 d0 i=1 i=1 + m (i) α(i) < a(i) , Wi ζ >< u1 , Wi ζ > − i=1 m (i) < Δ1 , Wi ζ > +μν2 + i=1 where: 2 β (i) = γ (i) (i) 2d0 (1 − a(i) 2 ) 2 ω (i) = γ (i) (i) 2d0 (2 + θ(i) α (i) (i) γ (i) (u0 − < a(i) , u1 >) (i) d0 (1 − a(i) 2 ) 2 (i) (i) (i) (i) (i) (i) (i) 2 − < u , a > −4(u − < u , a >) γ (i) u0 u (i) 1 0 1 = 1 + u1 2 + 0 (i) (i) 2 ) (1 − a d0 (i) 2 (i) (i) (u − < a(i) , u1 >)2 ) + 0 (1 − a(i) 2 ) (i) 2 α(i) = (i) u1 2 = φ(i) = ν1 = (i) (i) γ (i) (1 − u0 (u0 − < u1 , a(i) >)) γ 2 (i) (i) d0 m i=1 (i) d0 (1 − a(i) 2 ) (i)2 (i) u0 u1 2 + (i) (i) (u0 (u0 − (i) < u1 , a(i) >) (1 − a(i) 2 ) − 1) 2 φ(i) m (i) (i) (i) ν1 = θ < u1 , Wi ζ > +λ(i) < a(i) , Wi ζ > −Δ0 i=1 ν1 μ2 2 98 Kanuengkid, W. and Mouktonglang, T. Then we apply the first optimality of ρ(μ, ζ) to obtain the following: ∂ρ ν2 =0⇔μ=− ∂μ ν1 Consequently, we have ψ(ζ) = min{ρ(μ, ζ) : μ ∈ R} = (9) 2m+2 ζ, Mζ 1 lij (vi ⊗ vj )ζ > + < v0 , ζ > − + < ζ, 2 2 i,j=1 (i) 2 (Σm i=1 Δ0 ) 2ν1 where 2 M = γ (i) (i) w0 Wi∗ Wi √ vi = 2Wi∗ a(i) , f or i = 1, . . . , m √ (i) vm+i = 2Wi∗ u1 , f or i = 1, . . . , m m 1 (i) ∗ (i) v2m+1 = √ λ Wi a ν1 i=1 m 1 (i) ∗ (i) v2m+2 = √ θ Wi u1 ν1 i=1 m m 1 (i) (i) v0 = √ Δ0 (v2m+1 + v2m+2 ) − Wi∗ Δ1 , ν1 i=1 i=1 L =lij , where lij = 0 except lij =β (i) , for i = 1, . . . , m, (i) l(m+i)(m+i) =ω , for i = 1, . . . , m, l(2m+1)2(m+1) = − 1 and l(2m+2)2(m+2) = −1, l(i)(m+i) =l(m+i)(i) = α(i) , 2 f or i = 1, . . . , m. From procedure above we have the following theorem. Theorem 3.1. Problem (8) is equivalent to following problem 2m+2 < ζ, Mζ > 1 ψ(ζ) = lij < vi , ζ >< vj , ζ > + < v0 , ζ >→ min (10) + 2 2 i,j=1 ζ∈Z Proof. By a direct calculation of the procedure above . 99 KSH LQ-control problem Then to obtain KSH-direction, it suffices to solve the problem (10). The following theorem further simplifies the problem(10). Theorem 3.2. Let ζ0 be the optimal solution to the problem : < ζ, Mζ > + < v0 , ζ > 2 ζ∈Z (11) and ζi i = 1, . . . , 2m + 2 be the optimal solution to the problem : < ζ, Mζ > + < Vi , ζ > 2 ζ∈Z (12) Let S = (sij ), sij =< vi , ζj >, i , j = 1, . . . , 2m+2. where Vi = m j=1 lij vj Then the set of optimal solution to the problem (10). is in one to one correspondence with the set of solution of the system of linear equations ⎡ ⎤ ⎡ ⎤ δ1 < v1 , ζ0 > ⎢ ⎥ ⎢ ⎥ .. (I − S) ⎣ ... ⎦ = ⎣ (13) ⎦ . δ2m+2 < v2m+2 , ζ0 > More precisely if (δ1 , . . . , δ2m+2 ) is a solution to equation (13), then ζ(δ) = ζ0 + 2m+2 δi ζi (14) i,=1 is the optimal solution to the problem (10) For a proof see [1,5]. ∂ρ = 0, μ = − νν21 . Let ζ be an optimal solution to equation(10) , ∂μ Therefore we obtain the primal descent direction ξ = Λ(μ, ζ) = ((μ, W1 ζ), . . . , (μ, Wmζ)), and a dual descent direction 1 1 η = P (n) 2 L(d)−1 P (n) 2 ξ − Δ. Note that the problem (10) is the a finite-rank perturbation of the linearquadratic control problem which can be handled by a similar technique as in NT-direction case. Nevertheless, one of the drawbacks of KSH-direction is 100 Kanuengkid, W. and Mouktonglang, T. that in order to solve (10), it is require to solve 2m + 3 linear quadratic control problems and a (2m + 2) × (2m + 2) linear algebraic equations. By comparing to NT-direction case, solving the arose finite-rank perturbation of the linearquadratic control problem can be solved by solving m + 2 linear quadratic control problems and a (m + 1) × (m + 1) linear algebraic equations. However, the scaling points are not required for KSH-direction. 4 Numerical results We use the same test problems as in [2]. As it is shown in the table below, the duality gap decreases as the number of iterations increases. Our approximated solution approaches to the optimal solution according to the theory of interior-point method. However, when compare to NT-direction, the number of iterations for NT-direction is less than KSH-direction for the same tolerance. As for average time per iteration, NT-direction gains advantage for a large number of targets and that is because the system of algebraic equations obtained in KSH-direction is much larger than the system of algebraic equations in NT-direction. Due to the independence of scaling point for KSH-direction, however, our algorithm still works even when the scaling point reaches the boundary. T 1 2 3 4 5 Opt V alue 2035 2071 2075 2076 2077 Number of Iterations via NT-direction 6 6 6 6 6 Number of Iterations via KSH-direction 10 11 12 12 12 T 6 7 8 9 10 Opt V alue 2077 2077 2077 2077 2077 Number of Iterations via NT-direction 6 6 6 6 6 Number of Iterations via KSH-direction 13 13 13 13 13 5 Conclusion We describe an implementation of an infinite dimension interior point method for solving multi-criteria linear -quadratic control problem based on KSHdirection. It is shown that such a descent direction is independent on the scaling point. Each iteration requires solving finite rank perturbation of linearquadratic control problems and a system of algebraic equations. It is require KSH LQ-control problem 101 to solve 2m + 3 linear quadratic problems and a (2m + 2) × (2m + 2) linear algebraic equations. However, scaling points are not required for KSH-direction. Numerical results show rapid convergence to the optimal solution. Acknowledgement: This research was supported by the Commission for Higher Education, CHE, RG2550 Methods of Nonlinear Analysis. References [1] Faybusovich, L. and Tsuchiya, T., Primal-dual algorithm and infinitedimensional Jordan algebras of finite rank. New trend in optimization and computation algorithms (NTOC 2001, Kyoto). Mathematical Programming, Series B, 97(3), 471-493, 2003. [2] Faybusovich, L., Mouktonglang, T. and Tsuchiya, T., Implementation of infinite-dimensional interior-point method for solving multi-criteria quadratic control problem. Optimization Method and Software, 21:2, 315341, 2005. [3] Tsuchiya, T., Aconvergence analysis of the Scaling-invariant PrimaldualPath-following Algorithms for Second-order cone Programming. Optimization Method and Software,10/10,141-182, 1999. [4] Kojima, M., Shida, M. and Shindoh, S., Anote on the Nestorov-Todd and the Kojima-Shindoh-Hara serch direction in semidefinite programming. Optimization Method and Software,11/12, 47-52, 1999. [5] Faybusovich, L. and Mouktonglang, T., Finite rank perturbation of the linear-quadratic control problem. Proceeding of the American Control Conference, Denver, Colorado, June 4-6, pp. 5347-5350, 2003. Received: August, 2010