1. A subject information acquisition apparatus comprising: an observation system that observes a subject and obtains observation data; a memory that stores data representing characteristics of the observation system; and a processor coupled to the memory, wherein the processor is configured to: (i) set an underdetermined system of linear equations Ax=b, where A denotes a matrix of the data representing the characteristics of the observation system, b denotes a constant vector which is data obtained by the observation system, and x denotes unknown vector which is data representing information relating to an inside of the subject; (ii) set the vector x or a vector w obtained by w=Φx, where Φ is a conversion matrix, as a variable to be determined, in which the number of elements of a variable to be determined is N and the number of linear equations is M, where M and N are integers satisfying 1≦M<N; (iii) execute, in each iteration: an extraction step of extracting at least M non-zero element candidates from said N elements of said variable; a subproblem solving step of setting a subproblem having only said extracted non-zero element candidates as a variable, and solving said subproblem as an ill-posed problem; an updating step of updating values of said N elements of said variable to be determined on the basis of values of said non-zero element candidates obtained in said subproblem solving step; and a step of determining whether or not the iterative calculations are complete; (iv) determine a value at which a value of an objective function defined as a function of the variable or a residual norm of the system of linear equations is minimized, from among values of the variable obtained in each iteration, as a solution of said system of linear equations, when the iterative calculations are complete; and (v) generate data representing information relating to the inside of the subject on the basis of the determined solution.