So for each $i$ and $k$,Įach major 3-by-3 grid has exactly one number of each type (1 through 9). In equations, for each $j$ and $k$, we haveĮach column has exactly one number of each value, too. What are the other rules of Sudoku? Each row has exactly one number of each value from 1 to 9. So if x(i,j,k) represents the value of the (i,j) grid at level k, then we must have Of course, for this to make sense, the solution needs to have exactly one 1 in the stack at each grid coordinate. In other words, the (i,j) grid can be viewed as a stack of 9 levels, and a value k in the (i,j) grid becomes a 1 at stack level k. A clue of, say, 5 in the original (3,4) grid becomes a 1 at level 5 in the cube at grid coordinate (3,4). Think of the cube as a stack of 9-by-9 grids, one at level 1, one at level 2, up to a grid at level 9. The key trick in turning the rules of Sudoku into a binary integer program is to turn the 9-by-9 grid of clues and answers into a 9-by-9-by-9 cube of binary numbers. How to Model Sudoku As a Binary Integer Program sudokuEngine uses a binary integer programming formulation. We just have to figure out how to represent the rules of Sudoku as an integer linear programming problem. But we don't have to figure out an algorithm. Those all use ingenuity to develop a Sudoku-solving algorithm. Compare this with several Sudoku solvers in the File Exchange and Cleve's article about how to program a Sudoku solver. Isn't that great? You don't have to figure out a solution algorithm. How does sudokuEngine solve Sudoku puzzles? It simply encodes the rules and clues in a binary integer programming problem. Intcon variables are integer within tolerance, options.TolInteger = 1e-05 (the Tolerance of the optimal value options.TolGapAbs = 0 (the default value). Intlinprog stopped at the root node because the objective value is within a gap LP: Optimal objective value is 29565.000000. The documentation shows how to solve this Sudoku puzzle using the sudokuEngine function: drawSudoku(sudokuEngine(B)) Both the sudokuEngine and drawSudoku functions accept clues in this form, or the more conventional 9-by-9 matrix form. If the first row is, then the clue at row 1, column 2 is equal to 2. The clues are encoded as follows: each row of a clue matrix has the form. As you probably know, a Sudoku puzzle is to fill a 9-by-9 grid with integers from 1 through 9 so that each integer appears only once in each row, column, and major 3-by-3 square, as in the following diagram, which includes puzzle clues. Include and Solve a Hyper Sudoku Puzzle.How to Model Sudoku As a Binary Integer Program.You might not be familiar with Hyper Sudoku I will describe it presently. ![]() Then it extends the method to solving Hyper Sudoku puzzles. This article starts with an explanation of how the Sudoku puzzle solver works. There are examples in the documentation of a complicated factory production problem and solution, a travelling salesman problem, and a Sudoku puzzle solver. The addition of integer constraints opens a surprisingly wide field of problems that are now solvable with Optimization Toolbox. The intlinprog solver attempts to solve problems with linear objective functions, linear constraints, and (this is the new part) the constraint that some variables must be integer-valued. Beginning with Release 2014a, Optimization Toolbox has mixed-integer linear programming. I am excited to describe how to use a new solver. While I have not written a blog post for Loren before, if you use Optimization Toolbox™ or Global Optimization Toolbox then you might have read my work. Alan writes documentation for mathematical toolboxes here at MathWorks. The hybrid function option lets you improve a solution by applying a second solver after the first.I'd like to introduce this week's guest blogger Alan Weiss. You can use custom data types with the genetic algorithm and simulated annealing solvers to represent problems not easily expressed with standard data types. You can improve solver effectiveness by adjusting options and, for applicable solvers, customizing creation, update, and search functions. For problems with multiple objectives, you can identify a Pareto front using genetic algorithm or pattern search solvers. You can use these solvers for optimization problems where the objective or constraint function is continuous, discontinuous, stochastic, does not possess derivatives, or includes simulations or black-box functions. ![]() Toolbox solvers include surrogate, pattern search, genetic algorithm, particle swarm, simulated annealing, multistart, and global search. Global Optimization Toolbox provides functions that search for global solutions to problems that contain multiple maxima or minima.
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