# Properties of the Quadratic Transformation of Dual Variables
## 5 Test Problems
The resulting test problems are of the form (26)-(28) with the parameters \(c\), \(A\), \(b\), \(d\) determined by given: the number \(m\) of constraints, the dimension \(n\), the number \(mb\) of active constraints (27), the number of variables \(md\) lying on the upper boundary of the set \(Q\), and the number \(ma\) of variables lying on the lower boundary of the set \(Q\). Table 1 shows the form of the test functions \(f(x)\) and the corresponding names of the dual functions. A linear programming problem for the function \(f_{1}\) was generated. Since some of the constraints (27) may not be satisfied for known dual variables, a regularized function [34]\(f_{1}=\langle c,x\rangle+\alpha(x,x)/2\) was used to solve the generated problem, with \(\alpha=0.01\). The function \(f_{2}\) has the parameter \(\varepsilon=0.01\). Our functions are separable. Therefore, the solution of the problem \(x(y)=\underset{x\in Q}{\text{argmin}}L(x,y)\) necessary to obtain the dual function decomposes into a series of one-dimensional minimization problems. Consider the solution of the problem (21) with the function \(f(x,c)=f_{2}\). Variables \(u\) and \(y\) are related according to (20), which means  
\(\Phi(u)=\ \phi(y(u))=-\psi(y(u))=-\underset{x\in Q}{\text{min}}L(x,y(u))=- \underset{x\in Q}{\text{min}}[f(x,c)+\langle y(u),g(x)\rangle]=-\underset{x \in Q}{\text{min}}[\langle c,x\rangle+\frac{\varepsilon}{2}\sum\limits_{i=1}^ {n}x_{i}^{2}+\langle y(u),Ax-b\rangle]\)  
Under the condition that there are no constraints on the variable \(x\), having solved the system  
\(\frac{\partial L(x,y(u))}{\partial x_{i}}=c_{i}+\varepsilon x_{i}+\left[A^{T} y(u)\right]_{i}=0\)  
we obtain  
\(x_{i}^{*}=-\ \frac{\left[A^{T}y(u)+c\right]_{i}}{\varepsilon},\ i=1,2, \ldots,n\). (35)  
Given the constraints \(a\leq x\leq d\), the solution \(x(y(u))\) has the form  
\(x_{i}(y(u))=\left\{ \begin{array}{ll}a_{i},\ \text{if}x_{i}^{*}\leq a_{i}, \\ d_{i},\ \text{if}x_{i}^{*}\geq d_{i},\\ x_{i}^{*},\ \text{if}a_{i}<x_{i}^{*}<d_{i},\end{array} \right.\)\(i=1,2,\ldots,n\). (36)  
The parameters \(u\), \(x\), \(y\) resulting from the solution of the minimization problem are denoted by \(\hat{u}\), \(\hat{x}\), \(\hat{y}\), and the optimal parameters are \(u*\), \(x*\), \(y*\). Having solved the unconstrained minimization problem (21), we find the parameters \(\hat{u}\) and Lagrange multipliers \(\hat{g}=y(\hat{a})\). The solution of the primal problem follows from the Equations (35) and (36). For the dual function (5), the following relations  
\(f(x)\geq\psi(y)\), \(f(x*)=\psi(y*)\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**F(v,c)** & **\(\theta\)(u) (11)** & **\(\Phi\)(u) (31)** \\ \hline \(f_{1}=\langle c,x\rangle\) & LP\_R\_Y1 & LP\_R\_Y2 \\ \(f_{2}=\langle c,x\rangle+\varepsilon(x,x)/2\) & L2\_Y1 & L2\_Y2 \\ \(f_{3}=\sum_{i=1}^{n}c_{i}/x_{i}\) & 1/x\_Y1 & 1/x\_Y2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test functions and their names. are fulfilled for any admissible \(x\) and \(y\). Therefore, to assess the quality of the obtained solution, we will use the indicator  
\[\Delta\psi=(f(x*)-\psi(\hat{y}))/|f(x*)|. \tag{38}\]  
The second indicator will be obtained based on the assessment of the quality of the point \(\hat{x}\). The obtained solution of the primal problem may not satisfy the constraints (27). Therefore, the value of the objective function \(f(\hat{x})\) may turn out to be less than \(f(x*)\) not suitable. In order to evaluate the quality of the approximation of the minimum value, we need to obtain a solution in the admissible area. Taking into account the specifics of the test problems, we will make its rough correction by moving to the admissible region along the ray \(a-\hat{x}\), where the vector \(a\) is the lower bound of the set \(Q\). Let us denote \(v(\hat{\rho})=a+\beta\) (\(\hat{x}-a\)). We set \(x_{D}=v(\hat{\rho}_{D})\), where \(\beta_{D}\) is the maximum value \(\hat{\rho}\) at which the point satisfies the constraints (27).