Unconstrained inverse quadratic programming problem

The paper covers a formulation of the inverse quadratic programming problem in terms of unconstrained optimization where it is required to find the unknown parameters (the matrix of the quadratic form and the vector of the quasi-linear part of the quadratic form) provided that approximate estimates of the optimal solution of the direct problem and those of the target function to be minimized in the form of pairs of values lying in the corresponding neighborhoods are only known. The formulation of the inverse problem and its solution are based on the least squares method. In the explicit form the inverse problem solution has been derived in the form a system of linear equations. The parameters obtained can be used for reconstruction of the direct quadratic programming problem and determination of the optimal solution and the extreme value of the target function, which were not known formerly. It is possible this approach opens new ways in over applications, for example, in neurocomputing and quadric surfaces fitting. Simple numerical examples have been demonstrated. A scenario in the Octave/MATLAB programming language has been proposed for practical implementation of the method.

💡 Research Summary

The paper introduces a novel formulation of the inverse quadratic programming (IQP) problem that does not rely on explicit constraints or exact knowledge of the optimal solution and objective value of the forward problem. Instead, it assumes that only approximate pairs ((\hat x_i,\hat f_i)) are available, each lying in a small neighbourhood of the true optimal point (x^) and its corresponding function value (f(x^)). The forward problem is expressed in the standard quadratic form
\