Abstract:
A method and computer program product for optimization of large scale resource scheduling problems. Large scale resource scheduling problems are computationally very hard and extremely time consuming to solve. This invention provides a Lagrangian relaxation based solution method. The method has two distinct characteristics. First, the method is formal. It is completely structure-based and does not use any problem domain specific knowledge in the solution process, either in the dual optimization or the primal feasibility enforcement process. Second, updating the Lagrangian multipliers after solution of every sub-problem without using penalty factors results in fast and smooth convergence in the dual optimization. The combination of high quality dual solution and the structure-based primal feasibility enforcement produces a high quality primal solution with very small solution gap. An optimal solution is first found to the dual of the resource scheduling problem by sequentially finding a solution to a plurality of sub-problems and updating a set of values used in the dual problem formulation after each sub-problem solution is obtained. Coupling constraint violations are systematically reduced and the set of values are updated until a feasible solution to the primal resource scheduling problem is obtained. An initial set of multiplier values is further determined by solving a relaxed version of the primal problem where most of the local constraints except the variable bounds are relaxed.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention relates generally to large scale resource scheduling optimization, and more particularly, to general structure-based methods that are applicable to large scale resource scheduling optimization.  
       BACKGROUND OF THE INVENTION  
       [0002]     Resource scheduling is a special class of optimization problem, in which a mix of available resources are utilized to satisfy demand side requirements over a given time horizon at prescribed temporal or spatial resolutions. The decisions to be made involve the determination of optimal operation schedules for all resources involved. The operation schedule for a resource can be described by the startup and shutdown, and utilization at each time step over the scheduling horizon. The operation schedules for all the resources are determined such that various local and global constraints are satisfied and some generalized cost function is minimized. A well-known classic example of resource scheduling problem is the so called unit commitment problem in the electric power industry, where the resources are the thermal, hydro power plants in the system; the demand side requirements are the total customer hourly load over 24 or 168 hours. The startup, shutdown, and utilization of each plant is referred to as commitment, de-commitment. and dispatch. The operation schedule for each plant must satisfy the local constraints such as minimum up time, minimum down time, ramping constraint, and available capacity limitations. Collectively, all the plants committed must also satisfy global constraints such as various reserve requirements and energy balance constraints.  
         [0003]     The resource scheduling problem, by its combinatorial nature, is very hard. The computational burden increases exponentially with the number of resources and the time steps in the scheduling horizon. To overcome the “curse of dimensionality”, decomposition techniques based on Lagrangian relaxation theory are generally used. In these decomposition methods, the original optimization problem is relaxed by removing the so called “complicating constraints”, also known as “coupling constraints”, to obtain a separable optimization problem, which can be divided into many smaller independent optimization problems, usually referred to as sub-problems. All the sub-problems are solved, e.g., one sub-problem for each unit in the unit commitment problem, and the Lagrangian multipliers are updated at a high level. The solutions of the sub-problems are coordinated by a set of price signals for the complicating constraints.  
         [0004]     The performance of a resource scheduling method is measured by the following criteria: (1) optimality, (2) feasibility and (3) speed. Optimality measures how close the solution is to the theoretical best. A feasibility check ensures that all constraints have been satisfied. Speed is how fast the method is at finding the solution. The optimality of the solution is usually measured by the solution gap. The solution gap is a conservative estimate of the closeness of a solution to a theoretical optimum solution in terms of an objective value. Solution gap is defined by (OBJ−LB)/OBJ expressed as a percent, where OBJ is the best integer solution found and LB is the tightest lower bound known for the problem.  
         [0005]     An optimization method based on Lagrangian relaxation typically includes two main steps: (1) Lagrangian dual optimization to get an optimal dual solution; and (2) construction of a primal feasible solution. The problem encountered in the first step is an optimization of the non-differentiable dual function, which is commonly solved by variations of subgradient (SG) and cutting plane (CP) methods. This is an iterative process plagued by slow convergence, requiring hundreds, sometimes thousands of iterations. Quickly getting to the optimal dual solution is an attractive and challenging objective in the design of Lagrangian relaxation-based algorithms.  
         [0006]     The primal solution corresponding to the optimal dual solution in general is not ensured to be feasible for all the coupling constraints. The second step attempts to make the primal solution feasible by applying various problem dependent heuristics to adjust the primal solution until it becomes feasible. The heuristics rules are generally constructed from experience and insight into the specific problems at hand and lack generality. The method disclosed herein offers a huge performance advantage over the traditional Lagrangian relaxation-based methods by providing fast dual optimization and a problem independent process for feasibility enforcement.  
       SUMMARY OF THE INVENTION  
       [0007]     The present invention provides a complete solution process for large-scale resource scheduling optimization and has the following unique advantages: 
    1. the solution process is completely structure-based—being a formal method, the inventive solution is independent of problem domain knowledge and thus could be applicable to a wide class of problems;     2. the solution provides very fast dual optimization—obtaining a high quality dual solution in just a few iterations, the invention results in drastic execution speed improvement over other methods;     3. the solution uses structure-based heuristics to search for primal feasibility—no problem formulation specific heuristics are required, thus increasing reusability for other resource scheduling problems; and     4. the solution is a completely automated process.    
 
         [0012]     In an exemplary embodiment, the invention is directed to a method for optimization of large scale resource scheduling. An optimal solution is first found to the dual of the resource scheduling problem by sequentially finding a solution to a plurality of sub-problems and updating a set of multiplier values used in the dual problem formulation after each sub-problem solution is obtained. Lagrangian relaxation techniques can be used to obtain the dual solution and the set of values updated after each sub-problem is solved are Lagrangian multipliers. The solution to the dual optimization disclosed herein does not use any penalty factors. Coupling constraint violations are systematically reduced and the set of multiplier values are updated until a feasible solution to the primal resource scheduling problem is obtained. The method optionally includes estimating the lower bound to the primal resource scheduling problem which, in turn, is used as an estimate for the upper bound of the dual resource scheduling problem. An initial set of multiplier values is further determined by solving a relaxed version of the primal problem where most of the local constraints except the variable bounds are relaxed.  
         [0013]     The structure-based algorithm of the invention is particularly well-suited for large scale resource scheduling such as the unit commitment problem in the electric power industry wherein hydro and thermal generating units are scheduled for operation to meet demand over a period of time while minimizing total operating costs. In the simplest case, the scheduling problem is decomposed into a single sub-problem for each thermal plant and for each hydroelectric plant. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]     The invention is better understood by reading the following detailed description of the invention in conjunction with the accompanying drawings.  
         [0015]      FIG. 1  illustrates the processing logic for the structure-based algorithm for large scale resource scheduling optimization in accordance with an exemplary embodiment of the invention.  
         [0016]      FIG. 2  illustrates the processing logic for the problem setup step of the algorithm in an exemplary embodiment.  
         [0017]      FIG. 3  illustrates the processing logic for the lower bound and initial multiplier estimate step of the algorithm in an exemplary embodiment.  
         [0018]      FIG. 4  illustrates the processing logic for the initial sub-problem solution step of the algorithm in an exemplary embodiment.  
         [0019]      FIG. 5  illustrates the processing logic for the dual optimization step of the algorithm in an exemplary embodiment.  
         [0020]      FIG. 6  illustrates the processing logic for the primal feasible solution generation step of the algorithm in an exemplary embodiment. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0021]     The following description of the invention is provided as an enabling teaching of the invention in its best, currently known embodiment. Those skilled in the relevant art will recognize that many changes can be made to the embodiments described, while still obtaining the beneficial results of the present invention. It will also be apparent that some of the desired benefits of the present invention can be obtained by selecting some of the features of the present invention without utilizing other features. Accordingly, those who work in the art will recognize that many modifications and adaptations to the present invention are possible and may even be desirable in certain circumstances and are a part of the present invention. Thus, the following description is provided as illustrative of the principles of the present invention and not in limitation thereof, since the scope of the present invention is defined by the claims.  
         [0022]      FIG. 1  depicts the major steps in the inventive process in an exemplary embodiment. For convenience of description, and without any loss of generality, the resource scheduling problem is described herein as a minimization problem. It needs to be pointed out that the disclosed process is a meta-algorithm in the sense that it does not care how the specific sub-problems are formulated or solved, as long as the sub-problems can be solved by a suitable formulation and means. For example, the sub-problems could be expressed as linear programming problems or as non-linear programming problems. In the particular implementation realized to test the solution process, the sub-problems were formulated as mixed integer programming (MIP) and solved by a commercially available MIP solver.  
         [0023]     The first step (block  100 ) is the estimation of the lower bound of the problem to be solved. If the estimation is available from other sources, this step is optional. When the resource scheduling is formulated as a MIP problem, the lower bound can be quickly obtained by removal of sub-problem level constraints and integer relaxation. The resulting lower bound will be used as an estimate of the upper bound of the dual problem. If this step is executed, the multipliers for the complicating constraints can be estimated from the resulting linear programming (LP) problem.  
         [0024]     In the second step (block  102 ), all sub-problems are solved once for the initial set of multipliers to get the initial primal solution. If the first step is skipped, an initial multiplier set of zero can be used with very little effect on the convergence of the solution process.  
         [0025]     In the third step (block  104 ), the dual problem is optimized by sequentially solving sub-problems and updating the multipliers after each sub-problem solution. The Lagrangian dual function is evaluated after each sub-problem solution, and an update for the multipliers is calculated. This differs from the traditional approach where multipliers are updated after all sub-problems are solved for a given set of multipliers. This design also differs in fundamental ways from a prior art method that uses a set of penalty factors to reduce violation of the coupling constraints and requires some follow up process that uses problem dependent heuristics to make the primal solution feasible. The presence of penalty factors causes significant bias in the multiplier update process and reduces the optimality of the dual solution. The solution disclosed herein does not use any penalty factors in the dual optimization phase and does not concern itself with primal feasibility until the next step.  
         [0026]     The fourth step (block  106 ) searches for primal feasibility by systematically reducing violations of coupling constraints. It updates the multipliers based on the modified subgradients and solve the corresponding sub-problems until feasibility is achieved.  
         [0027]     The method successively solves resource scheduling problems with hundreds of thousands of variables and several hundred thousand constraints, producing solutions that are 100% feasible with very good solution gap, thereby reducing solution time by orders of magnitude as compared with optimization by a standard solver directly.  
         [0028]     For simplicity of notation, the inventive algorithm is described with respect to a reference model for resource scheduling optimization. The reference model is expressed as a mixed integer programming (MIP) problem, which is done solely to describe the principles of the invention.  
         [0000]     The Reference Model and Notations  
         [0000]     Minimize 
 
c T x 
 
 Ax=b  
 
 A   c   x&gt;=b   c  
 
 Subject to 
 
 B   l   &lt;=x&lt;=B   u  
 
x&gt;=0 
 
         [0029]     The above problem will be referred to as problem P. A c  is the constraint matrix for the coupling rows. A is the constraint matrix for the non-coupling rows. The inclusion of only greater than constraints for the coupling rows will not affect the generality of the method. By adding slack variables to the coupling rows, we get the following problem PS:  
         [0000]     Minimize 
 
 c   T   x+c   s   T   S   c  
 
 Ax=b  
 
 A   c   x+S   c   &gt;=b   c  
 
 Subject to 
 
 B   l   &lt;=x&lt;=B   u  
 
x&gt;=0 
 
         [0030]     The Lagrangian is defined as: 
 
 L (λ, x )= c   T   x+λ   T ( b   c   −A   c   x ) 
        where λ is Lagrangian multiplier vector for the coupling rows.        
 
         [0032]     The Lagrangian dual function is defined as:  
         L   ⁡     (   λ   )       =         Min   x     ⁢           ⁢     c   T     ⁢   x     +       λ   T     ⁡     (       b   c     -       A   c     ⁢   x       )             
 
 Ax=b  
 
 Subject to 
 
 B   l   &lt;=x&lt;=B   u  
 
x&gt;=0 
 
         [0033]     The Lagrangian dual function is separable into sub-problems. The k-th sub-problem, referred to as P(k), is defined by the following:  
           L   k     ⁡     (   λ   )       =         Min     x   k       ⁢           ⁢     c   k   T     ⁢     x   k       +       λ   T     ⁡     (       b   c     -         [     A   c     ]     k     ⁢     x   k         )             
 
[ A]   k   x   k   =b   k  
 
 Subject to 
 
 B   l   &lt;=x   k   &lt;=B   u  
 
 x   k &gt;=0 
        Where     x k  is the variable set for sub-problem k;     c k  is the coefficient set for sub-problem k;     [A c ] k  is the sub matrix of A c , made up of columns corresponding to sub-problem k;     [A] k  is the sub matrix of A, made up of columns corresponding to sub-problem k.        
 
         [0039]     The processing logic for the structure-based algorithm for optimization of large scale separable resource scheduling illustrated in  FIG. 1  does not show the initial step of problem setup. This initial step carries out the necessary preparation tasks for the succeeding steps. Several key problems and dual sub-problems are set up. Step  1  as indicated in logic block  100  obtains the estimation of the lower bound of the problem to be solved and the initial multipliers. All sub-problems are solved once for the initial set of multipliers to get the initial dual solution in the second step as indicated in logic block  102 . In step  3 , as indicated in logic block  104 , the dual problem is optimized by sequentially solving sub-problems and updating the multiplier after each sub-problem solution. In step  4 , as indicated in logic block  106 , primal feasibility is achieved by systematically reducing coupling constraints violations.  
         [0040]      FIG. 2  shows the detailed processing logic for the initial step of “problem setup”. In logic blocks  200  and  202 , the following key problems are set up: the original primal problem (P 0 ), the problem (P 1 ) obtained by cloning P 0  and adding slack variables, the relaxed problem (P 2 ) obtained by cloning P 0  and removing coupling constraints, and all the sub-problems of P 2  obtained by using given decomposition data. In logic block  204 , coupling row names can be directly obtained from other programs or read from external disk files. In logic block  206 , slack variables are added to the coupling rows of P 1  and appropriate objective coefficients are set for the slack variables. As illustrated later, P 1  will be used in the primal feasible solution generation step and the slack variables are used to calculate the infeasibility index. By deleting the coupling rows from P 2  (logic block  207 ), the relaxed problem is obtained, which can be decomposed into smaller and independent easier-to-solve sub-problems as indicated in logic block  208 . Decomposition is based on the constraint-to-sub-problem and variable-to-sub-problem mapping definitions, which specify the sub-problem to which each constraint and variable is partitioned. Accordingly, each of the sub-problems of P 2  is created and set up as indicated in logic block  210 .  
         [0041]      FIG. 3  illustrates the processing logic for the “lower bound and initial multiplier estimate” step of the algorithm. This step of the algorithm obtains the estimation of the lower bound of the problem to be solved. If the estimation is available from other sources, this step is optional. In an exemplary embodiment, the resource scheduling is formulated as a MILP problem, and the lower bound is obtained by removal of sub-problem level constraints and integer relaxation. Sub-problem level constraints may include unit minimum up and down time, and unit ramping up and down constraints when applied to a unit commitment problem in the electric power industry. Integer relaxation removes the integrality constraints on the problem (logic block  310 ), and makes the problem a Linear Programming problem (logic block  312 ). The resulting lower bound (logic block  316 ) will be used as an estimate of the upper bound of the dual problem. If this step of the algorithm is executed, the multipliers, i.e., the dual of the complicating constraints, can be readily obtained from the solution of the resulting linear programming problem (logic block  318 ). In logic block  302 , “get partial row names” obtains a list of string identifiers identifying the sub-problem level constraints to be deleted from the problem. The constraints with names that can match the string identifiers (logic block  306 ) will be removed from the problem (logic block  308 ).  
         [0042]     Pseudocode for the lower bound and initial multiplier estimate step of the algorithm is as follows:  
         [0000]     Lower Bound and Initial Multiplier Estimate Step (block  100 )  
         [0000]    
       
          Get problem P;  
          Remove local constraints;  
          Relax Integer variables;  
          Solve problem P;  
          Get objective value;  
          Get multipliers for coupling constraints.  
       
     
         [0049]      FIG. 4  illustrates the processing logic for the “initial sub-problem solution” step of the algorithm. In this step of the algorithm, all sub-problems are solved once for the initial set of multipliers to get the initial primal solution. If the “lower bound and initial multiplier estimation” step is skipped, an initial multiplier set of zero can be used (logic block  400 ), which will have little effect on the convergence of the solution process. As indicated in logic block  406 , given a set of multipliers, the objective coefficients of the sub-problem are updated as follows: 
   o   k   =c   k   T   x   k −λ T   [A   c ] k           where o k  are the objective coefficients of the sub-problem k, and λ is the given multiplier.          
         [0051]     Pseudocode for the initialization sub-problem solution for initial multiplier step of the algorithm is as follows:  
         [0000]     Initial Sub-Problem Solution for Initial Multipliers Step (Block  102 )  
         [0000]    
       
          Initialize multipliers λ;  
          Set Current Sub-problem Index k=0;  
          Set solution vector x=0;  
          While(k&lt;=NumOfSubproblems) 
        k=k+1;     Update sub-problem P(k) objective coefficients with current multipliers;     Solve current sub-problem;     Get solution for current sub-problem x k ;     Update solution x with x k ;    
     
          End While.  
       
     
         [0062]      FIG. 5  illustrates the processing logic for the “dual optimization” step of the algorithm. The dual problem is optimized by sequentially solving sub-problems and updating the multipliers after each sub-problem solution. The Lagrangian dual function is evaluated after each sub-problem solution (logic block  506 ), and an update for the multipliers is calculated (logic block  512 ). The subgradient g at the current solution is calculated as: 
   g=b   c   −A   c   x.    
         [0063]     The step size θ for multiplier update (logic block  510 ) is calculated as:  
         θ   =       (     LB   -     L   k       )            g        2         ,       
        where LB is the obtained estimate of the upper bound of the dual function, and L k  is the value of the dual function evaluated using the updated dual solution after solving the k th  sub-problem.        
 
         [0065]     The multiplier is updated (logic block  512 ) by using equation: 
 
λ new =λ old   +θ·g,  
        where λ old  and λ new  represent the old and new multipliers respectively.        
 
         [0067]     The objective coefficients of the current sub-problem are updated (logic block  514 ) in the same way as in the “initial sub-problem solution” step of the algorithm.  
         [0068]     Pseudocode for the dual optimization by sub-problem level dual function evaluation and multiplier update step of the algorithm is as follows:  
         [0000]     Dual Optimization by Sub-Problem Level Dual Function Evaluation and Multiplier Update Step (Block  104 )  
         [0000]    
       
          Set PassCount=0;  
          While(PassCount &lt;=MaxPassCount) 
        PassCount=PassCount+1;     Set Current Sub-problem Index k=0;     While(k&lt;=NumOfSubproblems) 
            k=k+1;     Evaluate Lagrangian function value at current multiplier and solution;     Calculate subgradient at current solution g=b c −A c x;     Calculate step size for multiplier update by  
         θ   =       (     LB   -     L   k       )            g        2         ;       
    Update multiplier by λ new =λ old +θ·g;     Solve sub-problem P(k) using new multipliers;     Get solution for current sub-problem x k ;     Update solution x with x k ;    
            End While;    
     
          End While. 
 
 Note: LB is the best estimate of the optimal value of the dual function. 
 
       
     
         [0084]      FIG. 6  illustrates the processing logic for the “primal feasible solution generation” step of the algorithm. This step of the algorithm searches for primal feasibility by systematically reducing coupling constraint violations. It proceeds as follows. First, the values of the integer variables of the dual solution are passed to the primal problem with slack variables (P 1 ) and fixed, and P 1  is solved to obtain a primal solution and the values of the slack variables. The infeasibility index is calculated as the sum of the absolute value of the slack variables. If the infeasibility index is zero, then a primal feasible solution is obtained and the program stops here; otherwise, it continues as follows. The subgradient is calculated as being equal to the value of the slack variable. Then the subgradient is projected to the positive quadrant as: 
 g=[g] + .  
         [0085]     The multiplier is updated as: 
 
λ new =λ old   +β·g,  
        where β is the selected step size. The recommended value for β is from 1 to 4. Smaller β generally results in a smaller solution gap and a greater number of iterations.        
 
         [0087]     Based on the new multiplier, objective coefficients of each of the sub-problems are updated similarly to step “initial sub-problem solution”, and the sub-problems are solved to obtain a new dual solution. Then pass the values of the integer variables of the dual solution to problem P 1 . This procedure is repeated until a primal feasible solution is achieved.  
         [0088]     Pseudocode for the primal feasible solution generation step of the algorithm is as follows:  
         [0000]     Primal Feasible Solution Generation Step (Block  106 )  
         [0000]    
       
          Set FeasibilityDone=false;  
          PassCount=0;  
          While(Not FeasibilityDone) 
        PassCount=PassCount+1;     Solve problem PS by fixing all integer variables based on the current solution x;     Calculate infeasibility index as sum of absolute value of slack variables;     If(infeasibility index=0) 
            FeasibilityDone=True;     Break;    
            Endif;     Calculate subgradient g=b c −A c x;     Project subgradient to the positive quadrant g=max(0, g)     Update multiplier based on the subgradient and selected step size λ new =λ old +β·g;     Set Current Sub-problem Index k=0     Set solution vector x=0;     While(k&lt;=NumOfSubproblems) 
            k=k+1;     Update sub-problem P(k) objective coefficients with current multipliers;     Solve current sub-problem; Get solution for current sub-problem x k ;     Update solution x with x k ;    
            End While;    
     
          End While.  
       
     
         [0111]     The method for optimization of large scale separable resource scheduling of the present invention has been described as a computer implemented process using application programs resident on a computer system that can process large scale optimization problems. It is important to note, however, that those skilled in the art will appreciate that the mechanisms of the present invention are capable of being distributed as a program product in a variety of forms, and that the present invention applies regardless of the particular type of signal bearing media utilized to carry out the distribution. Examples of signal bearing media include, without limitation, recordable-type media such as diskettes or CD ROMs, and transmission type media such as analog or digital communications links. Furthermore, those skilled in the art will appreciate that the broad concept of large scale resource scheduling optimization by sequential Lagrangian dual optimization and meta-heuristics to enforce a feasible solution are applicable to a wide range of computationally complex optimization problems and that the reference model described herein in the form of a mixed integer programming problem is used solely to describe an exemplary embodiment.  
         [0112]     Those skilled in the art will appreciate that many modifications to the preferred embodiment of the present invention are possible without departing from the spirit and scope of the present invention. In addition, it is possible to use some of the features of the present invention without the corresponding use of other features. Accordingly, the foregoing description of the preferred embodiment is provided for the purpose of illustrating the principles of the present invention and not in limitation thereof, since the scope of the present invention is defined solely by the appended claims.