Interior point

MacDonald, W. B., A. N. Hrymak, and S. Treiber. Interior Point Algorithms for Refinery Scheduhng Problems in Froc. 4th Annual Symp. Frocess Systems Engineering (Aug. 5-9, 1991) HI.13.1-16. 

Shock-compression processes are encountered when material bodies are subjected to rapid impulsive loading, whose time of load application is short compared to the time for the body to respond inertially. The inertial responses are stress pulses propagating through the body to communicate the presence of loads to interior points. In our everyday experience, such loadings are the result of impact or explosion. To the untrained observer, such events evoke an image of utter chaos and confusion. Nevertheless, what is experienced by the human senses are the rigid-body effects the time and pressure resolution are not sufficient to sense the wave phenomena. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

The result is that while there is, in DM, something that might be called an information cone centered at each site, it is not really what we usually think of as a relativistic, light cone, for wliidi we can point to interior points and definitely say they arc causally related and know for sure that points outside of each other s light cones are completely independent. In DM it is simply false to say that only those events inside the information cone of the past can influence a present event the information cone can well consist of lights cones stretching into all directions, forward and back in time. 

To know that a minimum exists, we must hnd three points < aj i < a ax such that F(ai t) is less than either F(a i ) or F a. Suppose this has been done. Now choose another point amin[Pg.207]

With these revised definitions for A, B, and C, the marching-ahead equation for the interior points is identical to that for cylindrical coordinates, Equation (8.25). The centerline equation is no longer a special case except for the symmetry boundary condition that forces a — 1) = a(l). The centerline equation is thus... 

Here, the temperatures on the left-hand side are the new, unknown values while that on the right is the previous, known value. Note that the heat sink/source term is evaluated at the previous location, — A. The computational template is backwards from that shown in Figure 8.2, and Equation (8.78) cannot be solved directly since there are three unknowns. However, if a version of Equation (8.78) is written for every interior point and if appropriate special forms are written for the centerline and wall, then as many equations are... 

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)]. 

As alternatives that avoid the combinatorial problem of selecting the active set, interior point (or barrier) methods modify the NLP problem (3-85) to form... 

LP software includes two related but fundamentally different kinds of programs. The first is solver software, which takes data specifying an LP or MILP as input, solves it, and returns the results. Solver software may contain one or more algorithms (simplex and interior point LP solvers and branch-and-bound methods for MILPs, which call an LP solver many times). Some LP solvers also include facilities for solving some types of nonlinear problems, usually quadratic programming problems (quadratic objective function, linear constraints see Section 8.3), or separable nonlinear problems, in which the objective or some constraint functions are a sum of nonlinear functions, each of a single variable, such as... 

Wright, S. J. Primal-Dual Interior-Point Methods. SIAM, Philadelphia, PA (1999). 

Sourander, M. L. M. Kolari J. C. Cugini J. B. Poje and D. C. White. Control and Optimization of Olefin-Cracking Heaters. Hydrocarbon Process, pp. 63-68 (June, 1984). Ye, Y. Interior Point Algorithms Theory and Analysis. Wiley, New York (1997). 

Bhatia, T. K. and L. T. Biegler. Multiperiod Design and Planning with Interior Point Methods. Comp Chem Engin 23 919 (1999). 

Y. Nesterov and A. S. Nemirovskii, Interior Point Polynomial Method in Convex Programming Theory and Applications, SIAM, Philadelphia, 1993. 

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