Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method

Abstract

Let <i >S</i> be a closed convex set of matrices and <i >C</i> be a given matrix. The matrix nearness problem considered in this paper is to find a matrix <i >X</i> in the set <i >S</i> at which max <svg style="vertical-align:-4.77652pt;width:71.987503px;" id="M1" height="16.700001" version="1.1" viewBox="0 0 71.987503 16.700001" width="71.987503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">{</tspan> <tspan style="font-size: 12.50px; " x="6.00144" y="0">|</tspan> <tspan style="font-size: 12.50px; " x="8.5020399" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,60.32)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-46.64,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="11.327718" y="0">𝑐</tspan> </text> <text transform="matrix(1,0,0,-1,-30.11,60.32)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-22.96,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">|</tspan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0">}</tspan> </text> </g> </g> </svg> reaches its minimum value. In order to solve the matrix nearness problem, the problem is reformulated to a min-max problem firstly, then the relationship between the min-max problem and a monotone linear variational inequality (LVI) is built. Since the matrix in the LVI problem has a special structure, a projection and contraction method is suggested to solve this LVI problem. Moreover, some implementing details of the method are presented in this paper. Finally, preliminary numerical results are reported, which show that this simple algorithm is promising for this matrix nearness problem.