Convex conditions definition

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

The first-order necessary conditions for problems with inequality constraints are called the Kuhn-Tucker conditions (also called Karush-Kuhn-Tucker conditions). The idea of a cone aids the understanding of the Kuhn-Tucker conditions (KTC). A cone is a set of points R such that, if x is in R, Tx is also in R for X 0. A convex cone is a cone that is a convex set. An example of a convex cone in two dimensions is shown in Figure 8.2. In two and three dimensions, the definition of a convex cone coincides with the usual meaning of the word. 

If the subspace S contains a positive definite element S, neither the energy problem nor the spectral optimization problem has an optimal solution since there is no positive semidefinite matrix P satisfying both the conditions (P, = 0, (P, I)j = 1, the convex set Pq n 5 is empty, and the energy pro-... 

Section 2.2 discusses the definitions and properties of convex and concave functions, the definitions of continuity, lower and upper semicontinuity of functions, the definitions of subgradients of convex and concave functions, the definitions and properties of differentiable convex and concave functions, the conditions of convexity and concavity along with their associated tests, and the definitions of extremum points. For further reading, refer to Avriel (1976), Mangasarian (1969), and Rockefellar (1970). 

The condition for stability of the ground state is that E[p] be a convex functional and therefore that rf r, r ), twice its Hessian, be positive-definite. r/ (r, r ) thus possesses a minimum eigenvalue and s (r, r ) a maximum, sj. A perturbation theoretic analysis sketched briefly in [3] then establishes that the upper bound of the spectrum of t]x(r, r ) is oo, and therefore the lower bound of the s is zero. 

The first and second conditions ensure the existence of the thermodynamic Gibbs free energy function or, using the mathematical term, the convex Lyapunov function for kinetic equations. The Lyapunov function is a strictly positive function with a nonpositive derivative. The one exception to this definition is that at the equilibrium point, the Lyapunov function equals zero. In physicochemical sciences, the Gibbs free energy is an extremely important Lyapunov function for understanding the stability of equilibria. 

One of the most successful QN formulas in practice is associated with the BFGS method (for its developers Broyden, Fletcher, Goldfard, and Shanno). The BFGS update matrix has rank 2 and inherent positive-definiteness (i.e., if B is positive definite then Bj + i is positive definite) as long as yjsk < 0. This condition is satisfied automatically for convex functions but may not hold in general. In practice, the line search must check for the descent property updates that do not satisfy this condition may be skipped. 

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