7.18 Support for Solving Inequalities

The operator ineq_solve tries to solves single inequalities and sets of coupled inequalities⁴ . The following types of systems are supported:

• only numeric coefficients (no parametric system),

• a linear system of mixed equations and \(<=\) – \(>=\) inequalities, applying the method of Fourier and Motzkin, as described by G. B. Dantzig in [Dan],

• a univariate inequality with \(<=\), \(>=\), \(>\) or \(<\) operator and polynomial or rational left–hand and right–hand sides, or a system of such inequalities with only one variable.

For linear optimization problems please use the operator simplex of the LINALG package (cf. section 20.32).

Syntax:

where \(<\)expr\(>\) is an inequality or a list of coupled inequalities and equations, and the optional argument \(<\)vl\(>\) is a single variable (kernel) or a list of variables (kernels). If not specified, they are extracted automatically from \(<\)expr\(>\). For multivariate input an explicit variable list specifies the elimination sequence: the last member is the most specific one.

An error message occurs if the input cannot be processed by the currently implemented algorithms.

The result is a list. It is empty if the system has no feasible solution. Otherwise the result presents the admissible ranges as set of equations where each variable is equated to one expression or to an interval. The most specific variable is the first one in the result list and each form contains only preceding variables (resolved form). The interval limits can be formal max or min expressions. Algebraic numbers are encoded as rounded number approximations.

Examples:

ineq_solve({(2*x^2+x-1)/(x-1) >=  (x+1/2)^2, x>0});

{x=(0 .. 0.326583),x=(1 .. 2.56777)}

 reg:=
 {a + b - c>=0, a - b + c>=0, - a + b + c>=0, 0>=0,
  2>=0, 2*c - 2>=0, a - b + c>=0, a + b - c>=0,
   - a + b + c - 2>=0, 2>=0, 0>=0, 2*b - 2>=0,
  k + 1>=0, - a - b - c + k>=0,
   - a - b - c + k + 2>=0, - 2*b + k>=0,
   - 2*c + k>=0, a + b + c - k>=0,
  2*b + 2*c - k - 2>=0, a + b + c - k>=0}$

ineq_solve (reg,{k,a,b,c});

{c=(1 .. infinity),

 b=(1 .. infinity),

 a=(max( - b + c,b - c) .. b + c - 2),

 k=a + b + c}

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