Best Approximate Solutions of Inconsistent Linear Inequality Systems

Approximate Solutions for Abstract Inequality Systems

We consider conic inequality systems of the type F (x) ≥K 0, with approximate solution x0 associated to a parameter τ , where F is a twice Fréchet differentiable function between Hilbert spaces X and Y , and ≥K is the partial order in Y defined by a nonempty convex (not necessarily closed) cone K ⊆ Y . We prove that, under the suitable conditions, the system F (x) ≥K 0 is solvable, and the rati...

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations

On the Approximation of Inconsistent Inequality Systems

In this paper is analyzed the minimal correction problem for an inconsistent linear inequality system. By the correction we mean avoiding its contradictory nature by means of relaxing the constraints. When the system of inequalities Ax ≤ b has no solutions, we are interested in a vector that satisfies the system in a Least Squares (LS) sense, i.e. a vector x ∈ R that minimizes the quantity ∥∥(A...

least squares solutions of inconsistent fuzzy linear matrix equations

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On Approximate Solutions of Systems of Linear Inequalities

Let ~x ~ b be a. consis tent system of linp,ar inequal ities. The principal result is a quantitative formulatIOn of the fa ct t hat if x "almost " sa t isfi es the inequalit ies t hen x is "close" to a solut ion. It is fur ther shown how i t is possible in certain cases to e~timate the $ize of the vector joining x to the nearest solu t ion from the magn itude of t he posit ive coordi nates of A...