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50-672: The CPLEX Optimizer was named for the simplex method implemented in the C programming language , although today it also supports other types of mathematical optimization and offers interfaces other than C. It was originally developed by Robert E. Bixby and sold commercially from 1988 by CPLEX Optimization Inc. This was acquired by ILOG in 1997 and ILOG was subsequently acquired by IBM in January 2009. CPLEX continues to be actively developed by IBM. The IBM ILOG CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of

100-401: A constraint equation is negative, the equation is negated before adding the identity matrix columns. This does not change the set of feasible solutions or the optimal solution, and it ensures that the slack variables will constitute an initial feasible solution. The new tableau is in canonical form but it is not equivalent to the original problem. So a new objective function, equal to the sum of

150-729: A program of repatriation of POWs after the Polish-Soviet War . He earned his Doctor of Philosophy degree at University of Warsaw in 1924 for a dissertation titled "On the Applications of the Theory of Probability to Agricultural Experiments". He was examined by Wacław Sierpiński and Stefan Mazurkiewicz , among others. He spent a couple of years in London and Paris on a fellowship to study statistics with Karl Pearson and Émile Borel . After his return to Poland, he established

200-537: A stand-alone Interactive Optimizer executable is provided for debugging and other purposes. The CPLEX Optimizer is accessible through independent modeling systems such as AIMMS , AMPL , GAMS , OptimJ and TOMLAB . In addition to that AMPL provides an interface to the CPLEX CP Optimizer. The full IBM ILOG CPLEX Optimization Studio consists of the CPLEX Optimizer for mathematical programming,

250-402: A vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself. Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized. Development of the simplex method

300-431: Is called pricing out and results in a canonical tableau where z B is the value of the objective function at the corresponding basic feasible solution. The updated coefficients, also known as relative cost coefficients , are the rates of change of the objective function with respect to the nonbasic variables. The geometrical operation of moving from a basic feasible solution to an adjacent basic feasible solution

350-571: Is defined by the constraints applied to the objective function. George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator . During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief , however, at that time he didn't include an objective as part of his formulation. Without an objective,

400-432: Is derived from the concept of a simplex and was suggested by T. S. Motzkin . Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones , and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope . The shape of this polytope

450-419: Is easily seen to be optimal since the objective row now corresponds to an equation of the form By changing the entering variable choice rule so that it selects a column where the entry in the objective row is negative, the algorithm is changed so that it finds the minimum of the objective function rather than the maximum. Once the pivot column has been selected, the choice of pivot row is largely determined by

500-409: Is implemented as a pivot operation . First, a nonzero pivot element is selected in a nonbasic column. The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the other rows to change the other entries in the column to 0. The result is that, if the pivot element is in a row r , then the column becomes the r -th column of

550-430: Is introduced with The second equation may be used to eliminate x 1 {\displaystyle x_{1}} from the linear program. In this way, all lower bound constraints may be changed to non-negativity restrictions. Second, for each remaining inequality constraint, a new variable, called a slack variable , is introduced to change the constraint to an equality constraint. This variable represents

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600-419: Is known as basic feasible solution (BFS). It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region, then it has this value on (at least) one of the extreme points. This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but

650-414: Is more than one column so that the entry in the objective row is positive then the choice of which one to add to the set of basic variables is somewhat arbitrary and several entering variable choice rules such as Devex algorithm have been developed. If all the entries in the objective row are less than or equal to 0 then no choice of entering variable can be made and the solution is in fact optimal. It

700-458: Is more than one row for which the minimum is achieved then a dropping variable choice rule can be used to make the determination. Consider the linear program With the addition of slack variables s and t , this is represented by the canonical tableau where columns 5 and 6 represent the basic variables s and t and the corresponding basic feasible solution is Columns 2, 3, and 4 can be selected as pivot columns, for this example column 4

750-406: Is no minimum. Next, the pivot row must be selected so that all the other basic variables remain positive. A calculation shows that this occurs when the resulting value of the entering variable is at a minimum. In other words, if the pivot column is c , then the pivot row r is chosen so that is the minimum over all r so that a rc > 0. This is called the minimum ratio test . If there

800-404: Is selected. The values of z resulting from the choice of rows 2 and 3 as pivot rows are 10/1 = 10 and 15/3 = 5 respectively. Of these the minimum is 5, so row 3 must be the pivot row. Performing the pivot produces Now columns 4 and 5 represent the basic variables z and s and the corresponding basic feasible solution is For the next step, there are no positive entries in

850-402: Is still in canonical form but with the set of basic variables changed by one element. Let a linear program be given by a canonical tableau. The simplex algorithm proceeds by performing successive pivot operations each of which give an improved basic feasible solution; the choice of pivot element at each step is largely determined by the requirement that this pivot improves the solution. Since

900-492: The Kamieniec Podolski gubernial gymnasium for boys in 1909 under the name Yuri Cheslavovich Neyman . He began studies at Kharkiv University in 1912, where he was taught by Ukrainian probabilist Sergei Natanovich Bernstein . After he read 'Lessons on the integration and the research of the primitive functions' by Henri Lebesgue , he was fascinated with measure and integration. In 1921 he returned to Poland in

950-783: The Royal Statistical Society on 19 June 1934, was the groundbreaking event leading to modern scientific sampling. He introduced the confidence interval in his paper in 1937. Another noted contribution is the Neyman–Pearson lemma , the basis of hypothesis testing. He was an Invited Speaker of the ICM in 1928 in Bologna and a Plenary Speaker of the ICM in 1954 in Amsterdam. In 1938 he moved to Berkeley, where he worked for

1000-714: The University of California, Berkeley . He was born into a Polish family in Bendery , in the Bessarabia Governorate of the Russian Empire , the fourth of four children of Czesław Spława-Neyman and Kazimiera Lutosławska. His family was Roman Catholic , and Neyman served as an altar boy during his early childhood. Later, Neyman would become an agnostic. Neyman's family descended from a long line of Polish nobles and military heroes. He graduated from

1050-537: The canonical form with c = ( c 1 , … , c n ) {\displaystyle \mathbf {c} =(c_{1},\,\dots ,\,c_{n})} the coefficients of the objective function, ( ⋅ ) T {\displaystyle (\cdot )^{\mathrm {T} }} is the matrix transpose , and x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\,\dots ,\,x_{n})} are

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1100-433: The feasible region defined by all values of x {\displaystyle \mathbf {x} } such that A x ≤ b {\textstyle A\mathbf {x} \leq \mathbf {b} } and ∀ i , x i ≥ 0 {\displaystyle \forall i,x_{i}\geq 0} is a (possibly unbounded) convex polytope . An extreme point or vertex of this polytope

1150-464: The simplex method or the barrier interior point method , convex and non-convex quadratic programming problems, and convex quadratically constrained problems (solved via second-order cone programming , or SOCP). The CPLEX Optimizer has a modeling layer called Concert that provides interfaces to the C++ , C# , and Java languages. There is a Python language interface based on the C interface. Finally,

1200-898: The Biometric Laboratory at the Nencki Institute of Experimental Biology in Warsaw. He published many books dealing with experiments and statistics, and devised the way which the FDA tests medicines today. Neyman proposed and studied randomized experiments in 1923. Furthermore, his paper "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection ", given at

1250-601: The CP Optimizer for constraint programming, the Optimization Programming Language (OPL), and a tightly integrated IDE. Prior to IBM acquiring ILOG, the CPLEX team published a release history of CPLEX. Simplex method In mathematical optimization , Dantzig 's simplex algorithm (or simplex method ) is a popular algorithm for linear programming . The name of the algorithm

1300-549: The artificial variables, is introduced and the simplex algorithm is applied to find the minimum; the modified linear program is called the Phase ;I problem. The simplex algorithm applied to the Phase I problem must terminate with a minimum value for the new objective function since, being the sum of nonnegative variables, its value is bounded below by 0. If the minimum is 0 then the artificial variables can be eliminated from

1350-405: The columns corresponding to the nonzero variables can be expanded to a nonsingular matrix. If the corresponding tableau is multiplied by the inverse of this matrix then the result is a tableau in canonical form. Let be a tableau in canonical form. Additional row-addition transformations can be applied to remove the coefficients c T B   from the objective function. This process

1400-407: The difference between the two sides of the inequality and is assumed to be non-negative. For example, the inequalities are replaced with It is much easier to perform algebraic manipulation on inequalities in this form. In inequalities where ≥ appears such as the second one, some authors refer to the variable introduced as a surplus variable . Third, each unrestricted variable is eliminated from

1450-401: The entering variable will, in general, increase from 0 to a positive number, the value of the objective function will decrease if the derivative of the objective function with respect to this variable is negative. Equivalently, the value of the objective function is increased if the pivot column is selected so that the corresponding entry in the objective row of the tableau is positive. If there

1500-451: The exact layout). If the columns of A {\displaystyle \mathbf {A} } can be rearranged so that it contains the identity matrix of order p {\displaystyle p} (the number of rows in A {\displaystyle \mathbf {A} } ) then the tableau is said to be in canonical form . The variables corresponding to the columns of the identity matrix are called basic variables while

1550-531: The existence of Lagrange multipliers for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed in the form of Lebesgue integrals . Dantzig later published his "homework" as a thesis to earn his doctorate. The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient. The simplex algorithm operates on linear programs in

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1600-429: The identity matrix. The variable for this column is now a basic variable, replacing the variable which corresponded to the r -th column of the identity matrix before the operation. In effect, the variable corresponding to the pivot column enters the set of basic variables and is called the entering variable , and the variable being replaced leaves the set of basic variables and is called the leaving variable . The tableau

1650-531: The initial identity matrix. However, the objective function W currently assumes that u and v are both 0. In order to adjust the objective function to be the correct value where u  = 10 and v  = 15, add the third and fourth rows to the first row giving Select column 5 as a pivot column, so the pivot row must be row 4, and the updated tableau is Jerzy Neyman Jerzy Neyman (April 16, 1894 – August 5, 1981; born Jerzy Spława-Neyman ; Polish: [ˈjɛʐɨ ˈspwava ˈnɛjman] )

1700-452: The linear program. When this process is complete the feasible region will be in the form It is also useful to assume that the rank of A {\displaystyle \mathbf {A} } is the number of rows. This results in no loss of generality since otherwise either the system A x = b {\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} } has redundant equations which can be dropped, or

1750-479: The linear program. This can be done in two ways, one is by solving for the variable in one of the equations in which it appears and then eliminating the variable by substitution. The other is to replace the variable with the difference of two restricted variables. For example, if z 1 {\displaystyle z_{1}} is unrestricted then write The equation may be used to eliminate z 1 {\displaystyle z_{1}} from

1800-453: The number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small. The solution of a linear program is accomplished in two steps. In the first step, known as Phase I, a starting extreme point is found. Depending on the nature of the program this may be trivial, but in general it can be solved by applying

1850-406: The objective function is unbounded above on the edge and the linear program has no solution. The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values. This continues until the maximum value is reached, or an unbounded edge is visited (concluding that the problem has no solution). The algorithm always terminates because

1900-468: The objective function is unbounded above. The transformation of a linear program to one in standard form may be accomplished as follows. First, for each variable with a lower bound other than 0, a new variable is introduced representing the difference between the variable and bound. The original variable can then be eliminated by substitution. For example, given the constraint a new variable, y 1 {\displaystyle y_{1}} ,

1950-410: The objective row and in fact so the minimum value of Z is −20. In general, a linear program will not be given in the canonical form and an equivalent canonical tableau must be found before the simplex algorithm can start. This can be accomplished by the introduction of artificial variables . Columns of the identity matrix are added as column vectors for these variables. If the b value for

2000-425: The original problem has no solution. Consider the linear program This is represented by the (non-canonical) tableau Introduce artificial variables u and v and objective function W  =  u  +  v , giving a new tableau The equation defining the original objective function is retained in anticipation of Phase II. By construction, u and v are both basic variables since they are part of

2050-424: The remaining columns with some other coefficients (these other variables represent our non-basic variables). By setting the values of the non-basic variables to zero we ensure in each row that the value of the variable represented by a 1 {\displaystyle 1} in its column is equal to the b {\displaystyle b} value at that row. Conversely, given a basic feasible solution,

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2100-808: The remaining variables are called nonbasic or free variables . If the values of the nonbasic variables are set to 0, then the values of the basic variables are easily obtained as entries in b {\displaystyle \mathbf {b} } and this solution is a basic feasible solution. The algebraic interpretation here is that the coefficients of the linear equation represented by each row are either 0 {\displaystyle 0} , 1 {\displaystyle 1} , or some other number. Each row will have 1 {\displaystyle 1} column with value 1 {\displaystyle 1} , p − 1 {\displaystyle p-1} columns with coefficients 0 {\displaystyle 0} , and

2150-418: The requirement that the resulting solution be feasible. First, only positive entries in the pivot column are considered since this guarantees that the value of the entering variable will be nonnegative. If there are no positive entries in the pivot column then the entering variable can take any non-negative value with the solution remaining feasible. In this case the objective function is unbounded below and there

2200-409: The resulting canonical tableau producing a canonical tableau equivalent to the original problem. The simplex algorithm can then be applied to find the solution; this step is called Phase II . If the minimum is positive then there is no feasible solution for the Phase I problem where the artificial variables are all zero. This implies that the feasible region for the original problem is empty, and so

2250-496: The simplex algorithm to a modified version of the original program. The possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called infeasible . In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point. The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which

2300-411: The smallest linear programs. It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the value of the objective function is strictly increasing on the edge moving away from the point. If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise

2350-449: The system is inconsistent and the linear program has no solution. A linear program in standard form can be represented as a tableau of the form The first row defines the objective function and the remaining rows specify the constraints. The zero in the first column represents the zero vector of the same dimension as the vector b {\displaystyle \mathbf {b} } (different authors use different conventions as to

2400-451: The variables of the problem, A {\displaystyle A} is a p × n matrix, and b = ( b 1 , … , b p ) {\displaystyle \mathbf {b} =(b_{1},\,\dots ,\,b_{p})} . There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality. In geometric terms,

2450-454: Was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson , revised Ronald Fisher 's null hypothesis testing. Neyman spent the first part of his professional career at various institutions in Warsaw , Poland , and then at University College London ; and the second part, at

2500-444: Was evolutionary and happened over a period of about a year. After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable. Dantzig realized that one of the unsolved problems that he had mistaken as homework in his professor Jerzy Neyman 's class (and actually later solved), was applicable to finding an algorithm for linear programs. This problem involved finding

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