# A Deterministic Dynamic Programming Approach for Optimization Problem with Quadratic Objective Function and Linear Constraints

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2 lower and upper limits of decision ith decision variable, respectively. 1 t n S 1 S t-1 S 0 Stage 1.. Stage t.. Stage n S t S n-1 S n Fig. 1 Multi-stage decision problem International Science Index, Mathematical and Computational Sciences waset.org/publication/17124 III. BACKWARD RECURSIVE APPROACH The optimum solution of n-variable problem is determined using dynamic programming by decomposing the original problem into n-stages. Each stage comprises a single variable sub-problem. Dynamic programming does not provide the computational details for optimizing each stage because the nature of the stage differs depending on the optimization problem. Such details are designed by the problem solver [10]. The computations in dynamic programming are carried out recursively that is the optimum solution of one sub-problem is taken as an input to the next sub-problem. The complete solution for the original problem is obtained by solving the last sub-problem. The sub-problems are linked together by some common constraints. The feasibility of constraints is accounted when each sub-problem is solved. Recursive equations are derived to solve the problem in sequence. These equations for multi-stage decision problems can be formulated in a forward or backward manner. Dynamic programming literature invariably uses backward recursion because, in general, it may be efficient computationally [11], [12]. A series of recursive equations are solved, each equation depending on the output values of the previous equation. Consider a multistage problem shown in Fig. 1. The stages 1, 2,, t,, n are labeled in an ascending order. For the t- stage, the input is denoted by s t-1 and output is denoted by s t. The objective of a multistage problem is to find optimal values of,,, so as to minimize the objective function subject to satisfying the equality and inequality constraints. The variable which links up two stages is called a state variable. The output of each stage is given by S 0, S S, S. Consider the last stage as the first sub-problem in the multistage decision process. The input variable for the last stage is s n-1. The relation between the state variables and decision variables are given by S S, 1,2,,. The quadratic objective optimization problem is solved by writing recursive equation for the last stage and then proceeding towards the first stage. The objective function of the last stage is given by S min (4) Now the last two stages are grouped together as the second sub-problem. The objective function is written as S min S (5) Substituting the objective function of nth stage in (5) gives S min (6) The output of the nth stage is given by The output of n-1 stage is given by Substituting (8) in (7) S S (7) S S (8) S S (9) Equation (6) is converted into a single variable problem by substituting (9) in (6), i.e, S min S S S S (10) The optimal value of variable x n-1 is obtained by differentiating (10) with respect to x n-1 and equating to zero, x n S S 0 (11) The expression for x n-1 is obtained from (11) S S (12) Substituting (12) in (9) gives the expression for the variable S S (13) By substituting the expressions for x n-1 and x n given in (12), (13) in (6), we get 566

3 S min 2 S S S S S S 2 2 S S (14) On simplifying, (14) becomes From (20), the expression for x n-2 is given by S S (21) In general, the optimal value of decision variable of the ith sub-problem is expressed from (21) International Science Index, Mathematical and Computational Sciences waset.org/publication/17124 S min (15) The output of the stage n-2 is given by S S (16) Substituting S S S S in (15) gives S mins S S S (17) The last three stages are grouped together as the third subproblem and the objective function is given by S min S (18) By substituting (17) in (18) gives S min S S S S (19) The optimal value of variable x n-2 is obtained by differentiating (19) with respect to x n-2 and equating to zero, 2 2S S 0 (20) S S ,2,, Using the generalized recursive given in (22), the solution of sub-problems from stage 1 to stage n-1 can be obtained and finally, the solution for nth stage is computed using (7). IV. COMPUTATIONAL PROCEDURE The procedure for implementing the proposed analytical approach for the solution quadratic objective function is detailed in the following steps: Step 1. Read the coefficients of the variables, lower and upper bounds of each variable and equality constraint of the given problem. Step 2. Treat the given optimization problem as a multistage problem. The problem is solved by breaking the original problem into a number of single stage problems. The number of single stage problem is equal to the number of variables. Step 3. Set S 0 =0, determine the optimal value of variable for the first stage problem using the generalized recursive equation given in (22). The optimal values of the remaining variables are determined using recursive equation by proceeding stage 1 to stage n in a forward manner. Step 4. During the recursive procedure if any variable violates their maximum or minimum limit then that variable is fixed at the corresponding violated limit and this subproblem is eliminated from the recursive procedure and this variable value is subtracted from the equality constraint. Now start the recursive procedure for stage 1. Step 5. Calculate the objective function of the given optimization problem using the optimal solutions obtained through the recursive procedure. V. COMPUTATIONAL STUDIES To test the performance of the proposed method, computational studies are performed for the real world optimization problem. The recursive approach is implemented 567

5 International Science Index, Mathematical and Computational Sciences waset.org/publication/17124 [2] O. Barrientos and R. Correa, An algorithm for global minimization of linearly constrained quadratic functions, Journal of global optimization, vol.16, pp , [3] R. E. Bellman, Dynamic programming, Princeton University Press, [4] V. Marano, G. Rizzo, and F. A. Tiano, Application of dynamic programming to the optimal management of a hybrid power plant with wind turbines, photovoltaic panels and compressed air energy storage, Applied Energy, vol. 97, pp , [5] I.R. Ilaboya, E. Atikpo, G. O. Ekoh, M. O. Ezugwu, and L. Umukoro, Application of dynamic programming to solving reservoir operational problems, Journal of applied technology in environmental sanitation, vol.1, pp , [6] John. O. S. Kennedy, Dynamic programming applications to agriculture and natural resources, Elsevier applied science publisher, [7] J. E. Beasley and B. C. Cao, A dynamic programming based algorithm for the crew scheduling problem, Computers & operation research, vol.25, pp , [8] R. Balamurugan and S. Subramanian, A simplified recursive approach to combined economic emission dispatch, Electric power components and systems, vol. 36, pp , [9] S. Webster and M. Azizoglu, Dynamic programming algorithms for scheduling parallel machines with family setup times, Computers & operation research, vol. 28, pp , [10] H. A. Taha, Operation research An introduction, Prentice- Hall India, [11] E. Denardo, Dynamic Programming theory and applications, Prentice Hall, [12] D. Bertsekas, Dynamic programming: Deterministic and stochastic models, Prentice Hall, [13] L. Bayon, J. M. Grau, M. M. Ruiz, and P. M. Suarez, The exact solution of the environmental/economic dispatch problem, IEEE transactions on power systems, vol. 27, pp , [14] D. P. Kothari and J. S. Dhillon, Power system optimization, Prentice Hall India, S. Kavitha received the B.Sc and M.Sc (Mathematics) degrees from Bharathidasan University, India, in 1995 and 1997, respectively. She completed M.Phil (Mathematics) from Madurai Kamaraj University, India. She is currently an Assistant Professor in Mathematics section, Faculty of Engineering and Technology, Annamalai University, India. Her research interests include optimization theory, soft computing and its applications. Nirmala. P. Ratchagar received the M.Sc (Applied Mathematics) from IIT, Kanpur and Ph.D from Kanpur University India. She has been involved in teaching for the past 22 years. She is currently a Professor in the Mathematics Department of Annamalai University. She has published twenty research papers in national and international journals. Her research topics include Mathematical modeling, fluid dynamics, optimization theory, computational intelligence and their applications. 569

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