Quadratic convex

These describing expressions are thus independent of knowledge of the crystallization kinetics. They imply that a quadratic convex form of controlled temperature profile is preferable to the more normal concave natural cooling curve. Further simplifications lead to a cubic cooling curve (Mullin and Nyvlt, 1971 Mayrhofer and Nyvlt, 1988). 

Given the pair and surface potentials, the weights are then constructed by solving the convex bound constrained quadratic program... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

Convex Cases of NLP Problems Linear programs and quadratic programs are special cases of (3-85) that allow for more efficient solution, based on application of KKT conditions (3-88) through (3-91). Because these are convex problems, any locally optimal solution is a global solution. In particular, if the objective and constraint functions in (3-85) are linear, then the following linear program (LP)... 

The parameter k in equation 67-1 induces a varying amount of nonlinearity in the curve. For the curves in Figure 67-1, k varied from 0 to 2 in steps of 0.2. The subtraction of the quadratic term in equation 67-1 gives the curves their characteristic of being convex upward, while adding the term kX back in ensures that all the curves, and the straight line, meet at zero and at unity. 

The result is illustrated in Figure 4.11 in which a convex quadratic function is cut by the plane /(x) = k. The convex set R projected on to the xt-x2 plane comprises the boundary ellipse plus its interior. 

For well-posed quadratic objective functions the contours always form a convex region for more general nonlinear functions, they do not (see tlje next section for an example). It is helpful to construct contour plots to assist in analyzing the performance of multivariable optimization techniques when applied to problems of two or three dimensions. Most computer libraries have contour plotting routines to generate the desired figures. 

EXAMPLE 6.3 APPLICATION OF NEWTON S METHOD TO A CONVEX QUADRATIC FUNCTION... 

Representation of the trust region to select the step length. Solid lines are contours of fix). Dashed lines are contours of the convex quadratic approximation of fix) at x. The dotted circle is the trust region boundary in which 8 is the step length. x0 is the minimum of the quadratic model for which H(x) is positive-definite. 

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. 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

Konno and Yamazaki (1991) proposed a large-scale portfolio optimization model based on mean-absolute deviation (MAD). This serves as an alternative measure of risk to the standard Markowitz s MV approach, which models risk by the variance of the rate of return of a portfolio, leading to a nonlinear convex quadratic programming (QP) problem. Although both measures are almost equivalent from a mathematical point-of-view, they are substantially different computationally in a few perspectives, as highlighted by Konno and Wijayanayake (2002) and Konno and Koshizuka (2005). In practice, MAD is used due to its computationally-attractive linear property. 

The exponential and logarithm functions have clear convex and concave characters, and do not do well in a nearly linear function. The last three functions provide minor corrections on the linear function, and do quite well. It should be pointed out that the first four functions have two arbitrary parameters each, the quadratic has three and the cubic has four. We expect that when the number of parameters increases, we can correlate ever more complicated data. For the cubic equation with three parameters and only six data points, there remain only three degrees of freedom. It is often said that Give me three parameters, and I can fit an elephant, so that it is a greater achievement to fit complicated data with as few parameters as possible, in the spirit of what is called the Occam s Razor. ... 

Let us consider the vapor pressure of water as a continuous function of temperature for the range of 0 to 100 °C. These data are also monotonic increasing and convex. Let us use as training set the data from 20 to 80 °C at 10 °C intervals, which is a set of seven points. Let us we propose a quadratic equation, and the regression result of... 

Illustration 3.2.4 Consider the following convex quadratic problem subject to a linear equality constraint ... 

The objective function /( ) and the inequality constraint g(x) are convex since f(x) is separable quadratic (sum of quadratic terms, each of which is a linear function of xi, x2,X3, respectively) and g(x) is linear. The equality constraint h(x) is linear. The primal problem is also stable since v(0) is finite and the additional stability condition (Lipschitz continuity-like) is satisfied since f(x) is well behaved and the constraints are linear. Hence, the conditions of the strong duality theorem are satisfied. This is why... 

Note that the objective function is convex since it has linear and positive quadratic terms. The only nonlinearities come from the equality constraint. By introducing three new variables w1,w2)w3, and three equalities ... 

Depending on the form of the objective function, the final formulation obtained by replacing the nonlinear Eq. (17) by the set of linear inequalities corresponds to a MINLP (nonlinear objective), to a MIQP (quadratic objective) or to a MILP (linear objective). For the cases where the objective function is linear, solution to global optimal solution is guaranteed using currently available software. The same holds true for the more general case where the objective function is a convex function. 

Usually the optimum lies on a convex or concave region of the response surface which can be approximated by functions with quadratic or cubic terms.)... 

Figure 5 illustrates more generally various cases that can occur for simple quadratic functions of form q x) — JxTHx, for n = 2, where H is a constant matrix. The contour plots display different characteristics when H is (a) positive-definite (elliptical contours with lowest function value at the center) and q is said to be a convex quadratic, (b) positive-semidefinite, (c) indefinite, or (d) negative-definite (elliptical contours with highest function value at the center), and q is a concave quadratic. For this figure, the following matrices are used for those different functions ... 

Contour plots for n = 2 also help illustrate paths toward minima specified by various minimization methods. The gradient vector is orthogonal to the contour lines. The familiar notion in one dimension that the negative tangent vector at a point x points toward the minimum of a convex quadratic extends naturally to higher dimensions. Thus, if the contour plots are circular... 

Convergence properties of most minimization algorithms are analyzed through their application to convex quadratic functions. A general multivariate convex quadratic can be written as... 

Steepest descent is simple to implement and requires modest storage, O(k) however, progress toward a minimum may be very slow, especially near a solution. The convergence rate of SD when applied to a convex quadratic function, as in Eq. [22], is only linear. The associated convergence ratio is no greater than [(k - 1)/(k + l)]4 where k, the condition number, is the ratio of largest to smallest eigenvalues of A ... 

The CG method was originally designed to minimize convex quadratic functions. Through several variations, it has also been extended to the general case.66-72... 

The first iteration in a CG method is the same as in SD, with a step along the current negative gradient vector. Successive directions are constructed differently so that they form a set of mutually conjugate vectors with respect to the (positive-definite) Hessian A of a general convex quadratic function. 

When one refers to the CG method, one often means the linear conjugate gradient, that is, the implementation for the convex quadratic form. In this case, minimizing IxTAx + bTx is equivalent to solving the linear system Ax = -b. Consequently, the conjugate directions pfe, as well as the lengths kh, can be computed in closed form. 

Truncated Newton methods were introduced in the early 1980s111-114 and have been gaining popularity ever since.82-109 110 115-123 Their basis is the following simple observation. An exact solution of the Newton equation at every step is unnecessary and computationally wasteful in the framework of a basic descent method. That is, an exact Newton search direction is unwarranted when the objective function is not well approximated by a convex quadratic and/or the initial point is distant from a solution. Any descent direction will suffice in that case. As a solution to the minimization problem is approached, the quadratic approximation may become more accurate, and more effort in solution of the Newton equation may be warranted. 

Summarizing At the time point t t-i the optimum of the [quadratic objective function] Zk is sought. The resulting control [input] vector U k) depends on x k—l) and contains all control [input] vectors u%, uf+i,. .., u% which control the process optimally over the interval tk-i, T. Of these control [input] vectors, one implements the vector (which depends on x k — 1)) as input vector for the next interval [t -i, tj. At the next time point a new input vector M +i is determined. This is calculated from the objective function Z +i and is dependent on x k). Therefore, the vector Ui, which is implemented in the interval is dependent on the state vector x k — 1). Hence, the sought feedback law consists of the solution of a convex optimization problem at each time point (k = 1, 2,. .., N). (Translation by the author.)... 

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