Convex functions Hessian matrix

Indefinite quadratic programs, in which the constraints are linear and the objective function is a quadratic function that is neither convex nor concave because its Hessian matrix is indefinite. 

A single-variable function is convex if the second derivative is strictly positive over the range of the dependent variable, as shown in Fig. 4(1). As show in Fig. 4(2), a function is concave when the second derivative is negative over the dependent variable range. For a multivariable function, the matrix of second derivatives (termed the Hessian, H(x)) is used to check the convexity (or concavity) of the function ... 

Since Dj = I, the initial iteration is a steepest descent step. Also, if / is a convex quadratic function, it can be shown that at termination Dj is the inverse of the Hessian matrix. 

Since Vf(x ) = 0, the algorithm terminates with x = (3, 2f. Note that Dj is precisely the inverse of the Hessian matrix, H(x) 2 convex quadratic objective function /(x). 

Nonlinear CG methods form another popular type of optimization scheme for large-scale problems where memory and computational performance are important considerations. These methods were first developed in the 1960s by combining the linear CG method (an iterative technique for solving linear systems Ax = b where A is an /i x /i matrix ) with line-search techniques. The basic idea is that if / were a convex quadratic function, the resulting nonlinear CG method would reduce to solving the Newton equations (equation 27) for the constant and positive-definite Hessian H. 

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