Second-order-necessary conditions

The second-order necessary conditions require this matrix to be positive-semidefinite on the tangent plane to the active constraints at (0,0), as defined in expression (8.32b). Here, this tangent plane is the set... 

Because — 2y is negative for all nonzero vectors in the set T, the second-order necessary condition is not satisfied, so (0, 0) is not a local minimum. 

The condition (3.4) is called the irst-order necessary condition the condition (3.7) is called the second-order necessary condition. 

Second-order necessary and sufficiency conditions for optimality... 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

In this section, we discuss the need for second-order optimality conditions, and present the second-order constraint qualification along with the second-order necessary optimality conditions for problem (3.3). 

Let it be a local optimum of problem (3.3), the functions f(x),h(x),g(x) be twice continuously differentiable, and the second-order constraint qualification holds at x. If there exist Lagrange multipliers A, fi satisfying the KKT first-order necessary conditions ... 

Illustration 3.2.12 This example is the motivating example for the second order necessary optimality conditions ... 

These boundary conditions are necessary and sufficient for the solution of equation 10.100 which is first order with respect to f and second order with respect to y. 

For the second-order difference equations capable of describing the basic mathematical-physics problems, boundary-value problems with additional conditions given at different points are more typical. For example, if we know the value for z = 0 and the value for i = N, the corresponding boundary-value problem can be formulated as follows it is necessary to find the solution yi, 0 < i < N, of problem (6) satisfying the boundary conditions... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

These investigators report that the second-order rate constant for reaction B is equal to 1.15 x 10 3 m3/mole-ksec at 20 °C. Determine the volume of plug flow reactor that would be necessary to achieve 40% conversion of the input butadiene assuming isothermal operating conditions and a liquid feed rate of 0.500 m3/ksec. The feed composition is as follows. 

Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function fix), given in Fig. 3-54, as a smooth valley in space of the variables X and x2. For the contour plot of this unconstrained problem Min/(x), consider a ball rolling in this valley to the lowest point offix), denoted by x. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector Vf(x) and second-derivative (Hessian) matrix V /(x) to state the necessary first- and second-order conditions for unconstrained optimality ... 

Second-Order Conditions As with unconstrained optimization, nonnegative (positive) curvature is necessary (sufficient) in all the allowable (i.e., constrained) nonzero directions p. The necessary second-order conditions can be stated as... 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Calculate the 99% confidence interval for predicting a single new value of response at pH = 7.0 for the data of Equation 11.16 and the second-order model of Equation 11.39. Calculate the 99% confidence interval for predicting the mean of seven new values of response for these conditions. Calculate the 99% confidence interval for predicting the true mean for these conditions. What confidence interval would be used if it were necessary to predict the true mean at several points in factor space ... 

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

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