LINEAR-QUADRATIC MODELS OF NONITERATIVE COORDINATION

LINEAR-QUADRATIC MODELS OF NONITERATIVE COORDINATION K.Stoilova, T. Stoilov Institute of Computer and Communication Systems – Bulgarian Academy of Sciences Sofia 1113, Acad. G.Bonchev str., bl.2, 979-27 74, 73-78-20, e-mail: k.stoilova@hsi.iccs.bas.bg, todor@hsi.iccs.bas.bg Abstract: Hierarchical approach is applied for solving linear-quadratic optimization problems. Non- iterative coordination in multilevel systems is considered. Explicit analytical relations are derived, concerning the problems of quadratic programming. Key words: Portfolio Optimisation, Two-Level Hierarchical Systems, Real Time Application 1. Introduction The reasons for organizing the control of large scale system in hierarchical manner refer to: decomposition of the complex system to a smaller subsystem, operating into hierarchical control structure [1]. The hierarchical control allows a reduction of the complexity in the control computations; easier way of system reaction to adopt to environmental changes, which implements robustness and scalability for the control. In general the control of a large scale system is performed and organized in a distributed hierarchy [2]. For example, in networked systems, information and commands are exchanged among multiple hosts and dependencies between these hosts change during operations. This challenge requires a shift in the focus of the control theory from prescribing to modelling and controlling the behaviour of the interacting distributed systems. Thus the control of distributed systems is organized in hierarchies in order to reduce the cognition and modelling efforts; the control and communication complexity. Through the hierarchy approach for the system design and control, it is expected: • To obtain modular and incremental design; • To increase the adaptation of the system towards changes in tasks and environmental influences; • To achieve a scalable design through decomposition and coordination of the large system components. The common approach in solving the problem of hierarchical control is to find optimal local controls influences among the subsystems in the hierarchical systems. The reasons for optimizing the exploitation of the system for efficiency purposes lead to the development of approaches deviating from the centralized design. Working with decomposition and coordination way, one is expecting benefits from the distributed and hierarchical system structure. Thus the control process is evaluated and is performed separately to each subsystem itself. Although there has been a lot of significant contributions in the area of large scale interconnected and multilevel systems, concerning system optimization and control, it can be said that the large scale systems do not constitute a complete and coherent theory. There is a lack of constructive and systematic methods and tools enabling to cope in a coherent way various hierarchical characteristics as multitime scale, multi-models, inexplicit mathematical relations, iterative and recursive calculations between system layers; different futures must be combined together with special decomposition, functional hierarchical distribution in order to achieve realistic real time and optimal management and resource allocation in large systems [3]. For that time, the hierachical control theory targeted the establishment of a new branch in Automatic Control with a new methodology in order to tackle in a realistic and efficient manner the distributed nature of the large systems. The achievements in state space domain, frequency domain, in the algebraic methodology seemed that all they could be put together, enlarged, adapted to propose new ways in dealing with distributed interconnected and hierarchical systems. The very structure of such systems, composed of subsystems with local input-output features and of an interconnection pattern, enabled the tendency in searching suitable design tools, benefiting from the interconnected and hierarchical feature. It is a real practical interest for such multilevel systems, because a great deal of important processes present such kind of interconnected structure. Hierarchical structure can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, game theory and economics. Moreover new applications are constantly being introduced. This stimulates the development of new theory and efficient algorithms. The implementation of the optimization methodology in the hierarchical control system resulted in the development of the multilevel optimization technique. Particularly for the two level hierarchical systems bi-level programming problems are defined and solved [4]. Thus the decision and control process in a hierarchy of control subsystems are defined and solved. Thus the decision and control process in a hierarchy of control subsystems are taken at different levels. In terms of modelling, the constraint domain associated with a multilevel programming problem is implicitly determined by a series of optimization problems, which must be solved in a predetermined sequence. [5]. The multilevel optimization technique is used to analyze decision made in hierarchical systems. The interrelationships among objectives and constraints of multiple levels result in a challenging mathematical problem. Without compromise or coordination, decisions in multilevel organizations can sometime be undesirable and inefficient, even when each level of the hierarchy operates optimally. The general multilevel programming problem is a set of nested optimization problems, over a single feasible region. Control over the decision variables is partitioned among the levels, but the decision variable may impact the objective function of several levels [6] [7]. The paper considers problems, motivated by bi-level optimization in hierarchical control structure. The optimization problems, solved by the subsystems are explicitly analytically derived. Non iterative coordination methodology is applied, which allows being derived explicit analytical solutions of the optimization subproblems. The bi-level modelling in the hierarchical system is done for the linear quadratic case. Duality theory is applied to the lower level problems. A minimax coordination problem is defined for which the optimality conditions are developed. Emphasis is given to the design of fast computational algorithms of bi-level optimization steady state problem. The research is directed towards development of no recursive and iterative manner of operation in the hierarchical system. This allows the deployment of non iterative coordination, performed by the upper level of the hierarchical system. The non iterative coordination decreases the inner intercommunication transfer between the subsystems [8]. 2. NONITERATIVE COORDINATION The coordination in a two level hierarchical system consists of iterative data transfer between the levels, fig.1. The coordinator defines a coordination parameter λ, which influences the subsystem optimization subproblems. With λ, the optimization subproblems Zi(λ), i=1,n, representing the subsystem management become well defined. For the given λ, the subsystems solutions are found. Next, the solutions xi(λ), i=1,n of Zi(λ), are sent back to the coordinator. The last, having evidence of the subsystem reactions xi(λ), i=1,n, improves the coordination from λ to λ* , λ*=λ*(xi(λ)), i=1,n, by means the local subsystems to find the global optimal solution. Next λ* is returned to the subsystem for implementation. When the coordination parameters λ influence the performance indices of the subsystems, this coordination is called “goal coordination”. If the coordination influences the constraints of the subsystems, the coordination is “predictive”. Thus an iterative communication-computing sequence is performed till finding optimal coordination λopt ,which results in optimal local solutions xiopt(λopt), i=1,n. Thus the multilevel system operates on optimal manner by solving a global optimization problem. The iterative coordination results in management delays, which don’t cope fast environmental changes for the hierarchical system. These delays are shorted with a new non-iterative coordination by reducing the information transfer between the system levels. The operation of the hierarchical system with non-iterative coordination strategy consists of two steps: - the coordinator sends to the subsystem initial coordination λ0; - using λ0 the subsystems solve their problems and evaluate the prepositions x(λ0); Coordinator Z1(λ λ x1(λ) λ x2(λ) λ Z2(λ ... xn(λ) Zn(λ) Fig.1. Two level hierarchical system - the coordinator corrects x(λ0) to the global optimal xopt or evaluate the optimal coordination λopt without iterative computations; the subsystems evaluate/implement x(λopt). This non-iterative concept can be applied both for “goal” and “interaction prediction” coordination strategies. 3. GOAL COORDINATION The global optimization problem, solved by the hierarchical system is stated in a separable form for the linear quadratic case as (1) min, F(x) = Error!x TQ x + RT x , g = A x x C=0 x = (x1, .., xN); QN x N; RN x 1; AM x N; CM x 1; g = (g1, .., gM), The multilevel theory founds on the mathematical modeling of the right and dual Lagrange problems and introduces the Lagrange parameters λ as coordination ones. According to the initial problem (1) the right Lagrange problem is stated as (2) min {L(x, λ)}, x L(x, λ) = 0.5 xTQ x + RTx + λT(A x–C), λ is a dual Lagrange multiplier. Problem (2) is an unconditional optimization one, which reduces to the linear equation system towards x as (3) ∂L = Q x + R + ATλ = 0. ∂x From (3) the solution x is expressed explicitly as a function of the coordination influence λ: (4) x(λ) = –Q–1(R + AT λ). The dual value λ must be evaluated from the Dual Lagrange problem stated as (5) max H(λ), λ ≥0 where H(λ)=L(x(λ), λ) is a non explicit analytical function. The last is defined from the Lagrange function L(x, λ) by substitution of the argument x with the function x(λ), established as a solution of the optimization problem (2). The function x(λ) is derived from (4). By substitution of (4) in (2) it follows explicit description of H(λ): H(λ)=L(x(λ), λ)=0.5xT(λ) Q x(λ) + RT x(λ) +λ T[A x(λ)–C] or after rearrangements it follows (6) H(λ)= –0.5λ TAQ–1AT λ–λ T(C+AQ–1R). Having H(λ) in explicit analytical way, the inverse Lagrange problem (5) reduces to unconstrained optimization, as a set of linear equations (7) max H(λ) , λ dλi dx dx d ( A) Ti = Q i Ti + AiT =0 dy iT dy i dy i dx i or dH ( λ ) = 0 = – AQ–1AT λ – (C + AQ–1R) and dλ λopt = – (AQ–1AT)–1(C + AQ–1R). (8) Ai Substituting of (8) in (4) explicit analytical solution of the initial optimization problem (1) is found xopt=–Q–1[R–AT(AQ–1AT)–1(C+A Q–1R)] . (9) From the first equation Thus the goal coordination applying the non iterative methodology gives explicit analytical solution (9) of the quadratic optimization problem (1). The analytical solution is a prerequisite for implementation real time control in large scales and hierarchical control systems. 4. NONITERATIVE ALLOCATION MODEL (22) A1 x1+ A2 x2 = T , R = R1 R2 ,x = x1 x2 Q= ( dx i = Q i−1 AiT Ai Q i−1 AiT dy iT ) −1 . With (20) and (22) the approximation of the inexplicit vector function xi(yi) , according to (21) becomes (21) RESOURCE ⎞ ⎛1 min ⎜ x T Qx + R T x ⎟ ⎠ ⎝2 dx i is expressed and then it is dy iT substituted in the second one and few rearrangements [ ] x i ( y i ) = − Q i R i − AiT ( Ai Q − 1 AiT ) − 1 Ai Q i− 1 R i + ( This model applies interaction-prediction coordination methodology, by means to solve the global optimization problem (18) dxi − I = 0. dyiT −1 + Q i−1 AiT Ai Q i−1 AiT ) −1 yi Here the unknown value is the resource yi , which can be found from the appropriate coordination problem. The last is defined by a substitution of the derived function xi(yi) in the initial optimization problem (18) or (24) min (w1 ( y1 ) + w 2 ( y 2 ) ) , y1 + y 2 = T , Q1 0 . 0Q2 y1 , y 2 To be applicable the predictive coordination approach problem (18) is modified in block-diagonal form where w ( y ) = 1 x T Q x + R T x or i i i i i i i ⎧⎛ 1 ⎞⎫ ⎞ ⎛1 min ⎨⎜ x1T Q1 x1 + R1T x1 ⎟ + ⎜ x2T Q2 x2 + R2T x2 ⎟⎬ 2 2 ⎠⎭ ⎠ ⎝ ⎩⎝ A1 x1 = y1 , A2 x2 = y2 , y1 + y 2 = T , which leads to the decomposition of two subproblems 2 1 T wi ( yi ) = yi ( Ai Qi−1 AiT ) −1 yi + RiT Qi−1 AiT ( Ai Qi−1 AiT ) −1 yi 2 1 T or wi ( yi ) = y i q i y i + ri T y i ,where 2 (19) (25) qi ⎞ ⎛1 min ⎜ x1T Q1 x1 + R1T x1 ⎟ , A1 x1 = y1 and ⎠ ⎝2 ⎛1 ⎞ min ⎜ x 2T Q 2 x 2 + R 2T x 2 ⎟ , A2 x2 = y2 . ⎝2 ⎠ [ riT = RiT Qi−1 AiT ( Ai Qi−1 AiT ) −1 . ⎧ 1 min⎨w( y) = y1 2 ⎩ Hence the coordination problem is ] The meaning of yi concerns the resources, allocated per subsystems. For yi=0 , the solutions of (19) from (9) is (20) xiopt 0 = −Qi Ri − AiT ( Ai Q −1 AiT ) −1 Ai Qi−1 Ri −1 To apply the noniterative methodology for problems (19) the inexplicit function xi(yi) must be approximated in Mac Loren series for the point (xi0, yi=0), or (21) x opt i ( y i ) = x i 0 + dx dy i T i .y x y The unknown matrices dxi dyi yi = 0 i 0 y i 0 , i 0 is calculated from the = ( Ai Qi−1 AiT ) −1 , (26) I y1 I y2 y2 ( q1 0 y1 0 q2 y 2 y1opt = −q1−1r1 + q1−1 q1−1 + q2−1 ) (q r2 T y1 ⎫ ⎬ y2 ⎭ r + q2−1r2 + T Using relation (9) the optimal resource allocation is opt i ( y2opt = −q2−1r2 + q2−1 q1−1 + q2−1 (27) −1 −1 1 1 −1 −1 1 1 ) (q r + q2−1r2 + T ) ) By a substitution of the resources у in (21),the optimal solution of the problem (18) is found in analytical form opt x1opt ( y1opt ) = −Q1−1 R1 + Q1−1 A1T ( A1Q1−1 A1T + A2 Q2−1 A2T ) × × ( A1Q1−1 R1 + A2 Q2−1 R2 + T ) (28) (A) × ( A1Q1−1R1 + A2Q2−1R2 + T ) By differentiating (А) and (В) towards y it holds T y1 =T . y2 appropriate dual and right Lagrange problems of (19), which reduces to the set of equation dLi dL = Qi xi + λTi Ai = 0 (B) Ti = Ai xi − yi = 0 T dxi dλi T T Li = 0 .5 x i Q i x i + λ i ( Ai x i − y t ) . + r1 x2opt ( y2opt ) = −Q2−1 R2 + Q2−1 A2T ( A1Q1−1 A1T + A2Q2−1 A2T ) × This an explicit analytical solution of (18). 5. EXAMPLE The initial optimization problem is given as 1 ⎧ ⎫ 2 2 2 2 ⎪ f ( x ) = ( 4 x1 + 3 x 2 + 2 x 3 + x 4 ) − ⎪ min ⎨ 2 ⎬ x ⎪⎩ − 6 x1 − 2 x 2 − 4 x 3 − 5 x 4 ⎪⎭ g 1 ≡ x1 + 2 x 2 + x 3 + 3 x 4 = 4 (29) g 2 ≡ 2 x1 + x 2 − 4 x 3 + x 4 = − 2 . The solution of this problem, using MATLAB is xT = (0,8851; opt − 0,1568; 0,2806; 1,0493) − 0 , 4897 ) . with duals λ Topt = (1, 4801 ; This problem is solved , applying the noniterative methodology with prediction coordination. The initial problem is decomposed to two subproblems with lower dimensions, which are solved in hierarchical order. First subproblem: ⎫ ⎧1 min f 1 ( x (1) ) = min ⎨ ( 4 x12 + 3 x 22 ) − 6 x1 − 2 x 2 ⎬ x1 , x 2 x1 , x 2 ⎭ ⎩2 (30) x (1) = x1 x2 g11 ≡ x1 + 2 x2 = y11 g12 ≡ −2 x1 + x2 = y12 , . ⎫ ⎧1 min f 2 ( x (1) ) = min ⎨ ( 2 x32 + x 42 ) − 4 x3 − 5 x 4 ⎬ x3 , x 4 x3 , x 4 ⎭ ⎩2 x3 x4 g 21 ≡ x3 + 3x4 = y21 g 22 ≡ −4 x3 + x4 = y 22 , λ (1 ) opt = ( a 1 q 1−1 a 1T ) −1 ( − a 1 q 1−1 r1 − c1 ) = with zero resources y2=0 is x0( 2 ) = r2 = −4 The decomposition of the constraints g1 and g2 of (29) is done accordingly g 1 = g 11 + g 21 = y11 + y 21 = 4 g 2 = g 12 + g 22 = y12 + y 22 = − 2 . The resources for the both subsystems are noted as −5 Evaluation of dx (1 ) dy 1T y=0 y =0 (31) x (1 ) ( y ) = 1 x ( 2) ( y2 ) = x3 x4 x1 x2 y 21 y 22 = x 0(1 ) + = x0( 2) dx 1 y 11 dy 12 y 12 dx 1 dy 11 dx2 + dy 21 dx2 y 21 dy 22 y 22 The explicit analytical description of the components (1) 0 x , ( 2) 0 x , dx (1) , dy1T dx ( 2 ) dy 2T ⎧1 min ⎨ y y11 y12 2 11 y21 y22 ⎩ x (1) 0 with zero resources y1=0 is x (1) 0 1 2 0 4 0, −6 , ; c1 = . a1 = r1 = q1 = −2 1 0 0 3 −2 = q 2−1 a 2T ( a 2 q 2−1 a 2T ) −1 = 1,8462 − 0 ,5385 0,2 − 0,4 0,4 0,2 − 0, 2308 0,0769 0,0769 0,3077 0,2 y11 − 0,4 y12 0,4 y11 + 0,2 y12 0,0769 y 21 − 0,2308 y 22 0,3077 y 21 + 0,769 y 22 1 + rr + y21 y12 qq1 y12 y12 2 0,64 { Applying problem. y11 − 0,08 T 1 . y11 (24) it y ⎫ + rr 21 ⎬ y22 qq2 y22 y22 ⎭ y21 y12 + y 22 = − 2 , where − 0,08 , 0,76 holds T 2 rr 1T = − 2 2 − 0,0118 rr2T = −1,8462 0,5385 0,1124 0,1065 − 0,0118 } min 0.5 y T QCy + RCy ACy = cc , where − 0 , 08 0 , 64 − 0 , 08 QC = 0 T AC = = −2 1 0 0 1 0 , 76 0 0 y T = y11 , because only the point (0,0) is admissible one. The corresponding parameters of this subproblem are . 0 The coordination problem is a quadratic programming one RC 0 = 0 0 = q 1−1 a 1T ( a 1 q 1−1 a 1T ) −1 = y 11 + y 21 = 4 follow: Evaluation of x0 . The solution of the first subproblem ;c = 2 3 −4 1 Coordination y . 1 dx . According to (22) it holds dy (32) x (1) ( y ) = 1 qq 2 = y2 = x0( 2 ) Thus the explicit description of the relations x(y) are: are expressed in Mac Loren series up to linear terms y12 and . 0 2 0 , and q2 = 0 0 1 λ ( 2 ) opt = ( a 2 q 2−1 a 2T ) −1 ( − a 2 q 2−1 r2 − c 2 ) = qq1 = y11 −2 The optimal dual for the second .The subproblem solutions y1 = ,a = 2 x ( 2) ( y 2 ) = . 2 Respectivly the solution of the second subsystem dx ( 2 ) dy 2T Second subproblem: x ( 2) = The optimal duals are 0 0 0 0 − 0 , 0118 0 ,1065 − 0 , 0118 2 − 1,8462 1 0 0 ; 1 y12 0 0 ,5385 ; CC = y 21 [ 0 ,1124 4 −2 ; y 22 ] The solution of the coordination problem is found due to the analytical relation (9) y1opt = − qq 1−1 rr1 + qq 1−1 qq 1−1 + qq 2−1 [ × qq 1−1 rr1 + qq 2−1 rr2 + T ] −1 × , or y opt = 0 , 5713 1 − 1, 9269 and y opt = 2 3 , 4286 . − 0 , 0729 The optimal resource allocations yopt are substituted in (32) and the final optimal solutions of the initial optimization program are x (1 ) opt = x ( 2 ) opt = x1 x2 x3 x4 = = 0 , 2 y 11opt − 0 , 4 y 12opt 0,4 y opt 11 + 0,2 y opt 12 = 0 ,8851 − 0 ,1568 opt opt 0,0769 y 21 − 0,2308 y 22 0,3077 y opt 21 + 0,0769 y opt 22 = 0,2806 1,0493 This solution is found without numerical iterative calculations, which speeds up the evaluation process. 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