An iterative approach to quadratic optimization. (English) Zbl 1043.90063

Xu, H. K.

J. Optimization Theory Appl. 116, No. 3, 659-678 (2003).

Summary: Assume that \(C_{1}, \dots , C_{N}\) are \(N\) closed convex subsets of a real Hilbert space \(H\) having a nonempty intersection \(C\). Assume also that each \(C_i\) is the fixed point set of a nonexpansive mapping \(T_i\) of \(H\). We devise an iterative algorithm which generates a sequence (\(x_n\)) from an arbitrary initial \(x_{0}{\in}H\). The sequence (\(x_n\)) is shown to converge in norm to the unique solution of the quadratic minimization problem \(min_{x\in C}(1/2){\langle}Ax, x{\rangle}-{\langle}x, u{\rangle}\), where \(A\) is a bounded linear strongly positive operator on \(H\) and \(u\) is a given point in \(H\). Quadratic-quadratic minimization problems are also discussed.

MSC:

90C20	Quadratic programming
90C48	Programming in abstract spaces

Keywords:

Iterative algorithms; quadratic optimization; nonexpansive mappings; convex feasibility problems; Hilbert spaces

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