InfSolve

Purpose

Find a constrained minimax solution with the use of any suitable TOMLAB solver.

infSolve 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 & \max r(x) \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\& c_L & \leq & c(x) & \leq & c_U \\ \end{array} }

where 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 x,x_L,x_U \in \mathbb{R}^{n}} , 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 r(x) \in \mathbb{R}^{N}} , 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 c(x),c_L,c_U \in \mathbb{R}^{m_1}} , 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 b_L,b_U \in \mathbb{R}^{m_2}} and 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 A \in \mathbb{R}^{m_2 \times n}} .

Calling Syntax

Result=infSolve(Prob,PriLev)

Inputs

Prob	Problem description structure. Should be created in the cls format. infSolve uses two special fields in Prob:
	SolverInf	Name of solver used to solve the transformed problem. Valid choices are conSolve, nlpSolve, sTrustr and clsSolve. If TOMLAB /SOL is installed: minos, snopt, npopt.
	InfType	1 - constrained formulation (default). 2 - LS penalty approach (experimental).
	The remaining fields of Prob should be defined as required by the selected subsolver.
PriLev	Print level in infSolve. =0 Silent except for error messages. >0 Print summary information about problem transformation. Calls PrintResult with specified PriLev. =2 Standard output from PrintResult (default).

Outputs

Result

Structure with results from optimization. Output depends on the solver used. The fields x_k, r_k, J_k, c_k, cJac, x_0, xState, cState, v_k are transformed back to match the original problem. g_k is calculated as J_kT · r_k. The output in Result.Prob is the result after infSolve transformed the problem, i.e. the altered Prob structure

Description

The minimax problem is solved in infSolve by rewriting the problem as a general constrained optimization problem. One additional variable 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 z\in \mathbb{R}} , stored as 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 x_{n+1}} is added and the problem is rewritten as:

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 z\\\\\mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n)^T & \leq & x_U \\& -\infty & \leq & z & \leq & \infty \\& b_L & \leq & A x & \leq & b_U \\& c_L & \leq & c(x) & \leq & c_U \\& -\infty & \leq & r(x) - z e & \leq & 0 \\\end{array} }

where 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 e \in \mathbb{R}^{N},\; e(i)=1 \ \forall i} .

To handle cases where an element 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 r_i(x)} in 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 r(x)} appears in absolute value: 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 \min \max |r_i(x)|} , expand the problem with extra residuals with the opposite sign: 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 [ r_i(x); -r_i(x) ] }

Examples

minimaxDemo.m.

See Also

clsAssign.