LinRatSolve
Purpose
Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables are not supported.
linRatSolve solves problems of the type:
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\ \end{array} }
where Failed to parse (unknown function "\MATHSET"): {\displaystyle c1,c2,x,x_L,x_U \in \MATHSET{R}^{n}}
, Failed to parse (unknown function "\MATHSET"): {\displaystyle b_L,b_U \in \MATHSET{R}^{m_1}}
and Failed to parse (unknown function "\MATHSET"): {\displaystyle A \in \MATHSET{R}^{m_1 \times n}}
.
Calling Syntax
Result=linRatSolve(Prob,PriLev)Inputs=
| Prob | Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored). | |
| PriLev | Print level in linRatSolve.
= 0 Silent except for error messages. > 0 Print summary information about problem transformation. Calls PrintResult with specified PriLev. = 2 Standard output from PrintResult (default). | |
| Extra fields used in Prob: | ||
| SolverRat | Name of the TOMLAB solver. Valid names are: cplex, minos, snopt, xa and more. See SolverList('lp'); | |
| QP.c1 | The numerator in the objective. | |
| QP.c2 | The denominator in the objective. | |
| z1_L | Lower bound for z1 (default 1e-5). See description below | |
Outputs
| Result | Structure with results from optimization. Output depends on the solver used.
The fields x_k, f_k, x_0, xState, bState, v_k are transformed back to match the original problem. The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure |
Description
The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.
The problem then becomes:
Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x c1 z2}\\\\\mbox{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\\end{array} }
where Failed to parse (unknown function "\MATHSET"): {\displaystyle $A1 \in \MATHSET{R}^{N},\; A1=speye(N)$} .
OBSERVE the denominator c2x must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).
