Publication date: 03/23/2021

Constrained Maximize(expr, {x1(low1, up1), x2(low2, up2), ...}, messages)

Description

Finds the values for the x arguments, specified as a list, that maximize the expr expression with optional linear constraints. You must either specify lower and upper bounds in parentheses for each argument or with the optional Set Variable Limit() message. The x arguments can be scalar values or vectors.

In the following messages, A is a matrix of coefficients. x = [x1, x2, ...] is the vector of arguments. b is a vector that forms the right side of the expression.

Messages

<<Less than EQ({A, b})

Sets the constraint to less than or equal to the specified values (A*x <= b).

<<Greater Than EQ({A, b})

Sets the constraint to greater than or equal to the specified values (A*x >= b).

<<Equal To({A, b})

Sets the constraint as equal to the specified values (A*x = b).

<<Starting Values([x1Start, x2Start, ...])

Specifies a starting point.

<<Max Iter(int)

An integer that specifies the maximum number of iterations to be performed.

<<Tolerance(p)

p sets the tolerance for the convergence criterion. The default tolerance is 10-5.

<<Show Details("true")

Returns a list with the final values for (objective value, number of iterations, gradient, and Hessian). Shows the step-by-step results of the optimizer in the log.

<<SetVariableLimit({low,high})

Specifies vectors for the lower and upper limits for the optimization variables.

Constrained Minimize(expr, {x1(low1, up1), x2(low2, up2), ...}, messages)

Description

Finds the values for the x arguments, specified as a list, that minimize the expr expression with optional linear constraints. You must either specify lower and upper bounds in parentheses for each argument or with the optional Set Variable Limit() message. The x arguments can be scalar values or vectors.

In the following messages, A is a matrix of coefficients. x = [x1, x2, ...] is the vector of arguments. b is a vector that forms the right side of the expression.

Messages

<<Less than EQ({A, b})

Sets the constraint to less than or equal to the specified values (A*x <= b).

<<Greater Than EQ({A, b})

Sets the constraint to greater than or equal to the specified values (A*x >= b).

<<Equal To({A, b})

Sets the constraint as equal to the specified values (A*x = b).

<<Starting Values([x1Start, x2Start, ...])

Specifies a starting point.

<<Max Iter(int)

An integer that specifies the maximum number of iterations to be performed.

<<Tolerance(p)

p sets the tolerance for the convergence criterion. The default tolerance is 10-5.

<<Show Details("true")

Returns a list with the final values for (objective value, number of iterations, gradient, and Hessian). Shows the step-by-step results of the optimizer in the log.

<<SetVariableLimit({low,high})

Specifies vectors for the lower and upper limits for the optimization variables.

Desirability(yVector, desireVector, y)

Description

Fits a function to go through the three points, suitable for defining the desirability of a set of response variables (y’s). yVector and desireVector are matrices with three values, corresponding to the three points defining the desirability function. The actual function depends on whether the desire values are in the shape of a larger-is-better, smaller-is-better, target, or antitarget.

Returns

The desirability function.

Arguments

yVector

Three input values.

desireVector

the corresponding three desirability values.

y

the value of which to calculate the desirability.

LPSolve(A, b, c, L, U, neq, nle, nge, <slackVars(Boolean)>)

Description

Returns a list containing the decision variables (and slack variables if applicable) in the first list item and the optimal objective function value (if one exists) in the second list item.

Arguments

A

A matrix of constraint coefficients.

b

A matrix that is a column of right hand side values of the constraints.

c

A vector of cost coefficients of the objective function.

L, U

Matrices of lower and upper bounds for the variables.

neq

The number of equality constraints.

nle

The number of less than or equal inequalities.

nge

The number of greater than or equal inequalities.

slackVars(Boolean)

(Optional) Determines whether the slack variables are returned in addition to the decision variables. The default value is 0.

Notes

The constraints must be listed as equalities first, less than or equal inequalities next, and greater than or equal inequalities last.

Maximize(expr, {x1(low1, up1), x2(low2, up2), ...}, messages)

Description

Finds the values for the x arguments, specified as a list, that maximize the expression expr. You can specify lower and upper bounds in parentheses for each argument. Additional arguments for the function enable you to set the maximum number of iterations, tolerance for convergence, and view more details about the optimization. The Newton-Raphson method is used when an analytical derivative is found for the Hessian. Otherwise, the Symmetric-Rank One method (SR1), a quasi-Newton method, is used.

Messages

<<Max Iter(int)

An integer that specifies the maximum number of iterations to be performed. The default maximum number of iterations is 250.

<<Tolerance(p)

p sets the tolerance for the convergence criterion. The default tolerance is 10-8.

<<Details("both" | "displaySteps" | "returnDetails")

Specifies what output is returned. If "displaySteps" is specified, step-by-step results of the optimization appear in the Log window. If "returnDetails" is specified, the function returns a list that contains the final values for the objective value, number of iterations, gradient, and Hessian. Specify "both" to get the return value and the results in the Log.

<<Gradient(exprList)

Specifies a list of expressions that define the analytical gradient that is used for the optimization. Each expression in the list represents a derivative of the expression expr.

<<Hessian(exprList)

Specifies a list of expressions that define the analytical Hessian that is used for the optimization. Each expression in the list represents the upper triangular portion of the Hessian matrix in row-major order.

<<Method(NR | SR1)

Specifies either the Newton-Raphson (NR) method or the Symmetric-Rank One (SR1) method for the optimization method.

<<UseNumericDeriv("true")

Specifies that the optimization use a numeric approximation.

Minimize(expr, {x1(low1, up1), x2(low2, up2), ...}, messages)

Description

Finds the values for the x arguments, specified as a list, that minimize the expression expr. You can specify lower and upper bounds in parentheses for each argument. Additional arguments for the function enable you to set the maximum number of iterations, tolerance for convergence, and view more details about the optimization. The Newton-Raphson method is used when an analytical derivative is found for the Hessian. Otherwise, the Symmetric-Rank One method (SR1), a quasi-Newton method, is used.

Messages

<<Max Iter(int)

An integer that specifies the maximum number of iterations to be performed. The default maximum number of iterations is 250.

<<Tolerance(p)

p sets the tolerance for the convergence criterion. The default tolerance is 10-8.

<<Details("both" | "displaySteps" | "returnDetails")

Specifies what output is returned. If "displaySteps" is specified, step-by-step results of the optimization appear in the Log window. If "returnDetails" is specified, the function returns a list that contains the final values for the objective value, number of iterations, gradient, and Hessian. Specify "both" to get the return value and the results in the Log.

<<Gradient(exprList)

Specifies a list of expressions that define the analytical gradient that is used for the optimization. Each expression in the list represents a derivative of the expression expr.

<<Hessian(exprList)

Specifies a list of expressions that define the analytical Hessian that is used for the optimization. Each expression in the list represents the upper triangular portion of the Hessian matrix in row-major order.

<<Method(NR | SR1)

Specifies either the Newton-Raphson (NR) method or the Symmetric-Rank One (SR1) method for the optimization method.

<<UseNumericDeriv("true")

Specifies that the optimization use a numeric approximation.

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