Convex computer

The work in Erlangen was supported by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the Convex Computer Corporation. The work in Georgia was supported by the U.S. Department of Energy, the U.S. National Science Foundation, and the U.S. Air Force Office of Scientific Research. 

The research in this area in Cambridge (and earlier in Strathclyde) has been supported by the U.K. Science and Engineering Research Council and by the Associated Octel Company Ltd. Organo-alkali metal research in Erlangen is supported by the Deutsche-forschungsgemeinschaft, the Fonds der Chemischen Industrie, Stiftung Volkswagen-werk, and the Convex Computer Corporation. Such financial input is essential, and it is appreciated. 

Calculations at the Zentrum fur Datenverarbeitung an der Universitat Tubingen using a Convex computer C3240 and the release H of GAUSSIAN 90185. 

If the prices used for P f, P, and Po are calculated assuming that the bond s remaining cash flows will not change when market rates do, the convexity computed is for an option-free bond. For bonds with embedded options, the prices used in the equation should be derived using a binomial model, in which the cash flows do change with interest rates. The result is effective or option-adjusted convexity. 

The most commercially successful of these systems has been the Convex series of computers. Ironically, these are traditional vector machines, with one to four processors and shared memory. Their Craylike characteristics were always a strong selling point. Interestingly, SCS, which marketed a minisupercomputer that was fully binary compatible with Cray, went out of business. Marketing appears to have played as much a role here as the inherent merits of the underlying architecture. 

MIMD Multicomputers. Probably the most widely available parallel computers are the shared-memory multiprocessor MIMD machines. Examples include the multiprocessor vector supercomputers, IBM mainframes, VAX minicomputers. Convex and AUiant rninisupercomputers, and SiUcon... 

Convex hull formulations of MILPs and MINLPs lead to relaxed problems that have much tighter lower bounds. This leads to the examination of far fewer nodes in the branch and bound tree. See Grossmann and Lee, Comput. Optim. Applic. 26 83 (2003) for more details. 

Both simulations were run on an HP Convex S-Class computer. aBased on a simulation on one processor. bConverged after 20,000 iterations. 

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. 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

Table 9.1 shows how outer approximation, as implemented in the DICOPT software, performs when applied to the process selection model in Example 9.3. Note that this model does not satisfy the convexity assumptions because its equality constraints are nonlinear. Still DICOPT does find the optimal solution at iteration 3. Note, however, that the optimal MILP objective value at iteration 3 is 1.446, which is not an upper bound on the optimal MINLP value of 1.923 because the convexity conditions are violated. Hence the normal termination condition that the difference between upper and lower bounds be less than some tolerance cannot be used, and DICOPT may fail to find an optimal solution. Computational experience on nonconvex problems has shown that retaining the best feasible solution found thus far, and stopping when the objective value of the NLP subproblem fails to improve, often leads to an optimal solution. DICOPT stopped in this example because the NLP solution at iteration 4 is worse (lower) than that at iteration 3. 

Westerlund, T. H. Skrifvars I. Harjunkoski, and R. Pom. An Extended Cutting Plane Method for a Class of Non-convex MINLP Problems. Comput Chem Eng 22 357-365 (1998). 

For zeolite structural units of the above size detailed ab initio calculations are prohibitively expensive even with the currently available most advanced computer programs. Convexity relation (13), and the resulting energy bounds, on the other hand, are easily applicable to a variety of similar problems, and require only few elementary algebraic operations. 

The actual program used at NPL was written by N.P. Barry on the basis of the methods described previously. It is written in FORTRAN and has been implemented on IBM 370 and UNIVAC 1100 computers operated by computer bureaux. Vector algebra is employed. The reason why the graphs have double boundaries is that the calculation can be performed for boundaries of any convex polygon of up to 30 sides. This permits calculations to be restricted to the stability range of particular components, for example, that of water or chloride. 

Our discussion may readily be extended from 2-positivity to p-positivity. The class of Hamiltonians in Eq. (70) may be expanded by permitting the G, operators to be sums of products of p creation and/or annihilation operators for p > 2. If the p-RDM satisfies the p-positivity conditions, then expectation values of this expanded class of Hamiltonians with respect to the p-RDM will be nonnegative, and a variational RDM method for this class will yield exact energies. Geometrically, the convex set of 2-RDMs from p-positivity conditions for p > 2 is contained within the convex set of 2-RDMs from 2-positivity conditions. In general, the p-positivity conditions imply the (7-positivity conditions, where q < p. As a function of p, experience implies that, for Hamiltonians with two-body interactions, the positivity conditions converge rapidly to a computationally sufficient set of representability conditions [17]. 

The 2-RDM/or the radical may be computed from the N + l)-electron density matrix for the dissociated molecule by integrating over the spatial orbital and spin associated with the hydrogen atom and then integrating over N — 2 electrons. Because the radical in the dissociated molecule can exist in a doublet state with its unpaired electron either up or down, that is, M = the 2-RDM for the radical is an arbitrary convex combination... 

Levy identified the unknown part of the exact universal D functional as the correlation energy Ed D] and investigated a number of properties of c[ D], including scaling, bounds, convexity, and asymptotic behavior [11]. He suggested approximate explicit forms for Ec[ D] for computational purposes as well. Redondo presented a density-matrix formulation of several ab initio methods [26]. His generalization of the HK theorem followed closely Levy s... 

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. 

Hippe et al. discussed numerical operations for computer processing of (gas) chromatographic data. Apart from a baseline correction method, a method of reco -tion of peaks is described. The relationship between the convexity of an isolated peak and the monotonic nature of its first derivative is used to find the most probable deflection points. The munber of maxima and shoulders are used for a decision if the segment of the chromatogram contains an isolated peak or an unresolved peak complex. The number of shouders and maxima determine the total number of component peaks. 

---