Convex optimization

A. Ben-Tal and A. Nemirovski, Lectures on Modem Convex Optimization Analysis, Algorithm, and Engineering Applications, SIAM, Philadelphia, 2(X)1. 

Case (a) may lead to parallel computations of the independent subproblems. Case (b) allows the use of special-purpose algorithms (e.g., generalized network algorithms), while case (c) invokes special structure from the convexity point of view that can be useful for the decomposition of non-convex optimization problems Floudas et al. (1989). 

Perrone, M. P. (1994). General averaging results for convex optimization. In Proceedings 1993 Connectionist Models Summer School (ed. Mozer, M. C., et al.), pp. 364-71. Lawrence Erlbaum, Hillsdale, NJ. 

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.)... 

There are several additional parameters in gradient analysis23 not present in isocratic HPLC that need to be optimized. These are initial and final mobile phase composition, gradient time or duration (tG), flow (F), and sometimes gradient curvature (linear, concave, and convex). Optimization of all these parameters is not intuitive but can often be readily accomplished by software simulation programs.2... 

Srirola J.D., Hauan S. and Westerberg A.W. 2003. Toward agent-based process systems engineering proposed framework and apphcation to non-convex optimization. Comp. Chem. Eng, 27, 1801-1811. 

Variational minimization of the energy as a functional of the 2-RDM is expressible as a special convex optimization problem known as a semidefinite program [33, 37, 41,42, 53, 58]. The convexity of the A-representable set of 2-RDMs ensures a rigorous lower bound to the ground-state energy. Because the variational 2-RDM method... 

The static problem considers the single-period analogue of the multiperiod problem in which the agent s utility is equal to his single-period reservation utility. This problem can then be used to construci the solution to the multiperiod problem as it will be explained below. The problem is formulated as follows For each state j = 1, and action a e A j) determine a set of compensation rates s jj a) k = 1, that solve the following convex optimization problem ... 

S. Boyd, L. Vandenberghe, Convex Optimization. Cambridge University, Cambridge, UK, 2004. 

Sapatnekar, S., Rao, V., and Vaidya, P. 1992. A convex optimization approach to transistor sizing for CMOS circuits. Proc. ICCAD, pp. 482-485. 

Though there exists a multitude of different possible problem statements, depending on the different technical tasks, a typical design optimization problem, with a combined mechanical (superscript m) and control subsystem (superscript c), is the following nonlinear (and usually non-convex) optimization problem ... 

J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo Publishing, Stanford, 2005. 

Particle Swarm Optimization (PSO) The optimization method based on the study of social behaviour in a self-organized population system (i.e., ant colonies, fish schools). Itisanon-gradient, heuristic method which requires calculation of the objective function only. This method is able to find a global solution to non-convex optimization problem and problems which have many local minima. 

In this section a scheme to derive the numerical value for the nonlinearity measure by convex optimization has been presented. The method is computationally efficient and standard tools for technical computing can be used. By an appropriate choice of the parameters and considered input signals, the error of the numerical approximation can be made arbitrarily small. 

Solution of Underestimating Problems. The parameters a,y r are updated for both rectangles r = a, b). The convex optimization problem (15) is solved inside both subrectangles (r = a, b) using a nonlinear solver (e.g., MINOS5.4 [27], NPSOL [28]). If a solution //y is less than the current upper bound, f, then it is stored. 

This problem, being the most expensive part of FORM algorithm, may be solved in a number of different ways (see (Eldred, Bichon, Adams 2006) for an overview). An appropriate iteration scheme converges after some iterations, providing the design point u as well as the rehabUity index J3, which may be related directly to the probability of failure. However, with non convex optimization problems, it is not guaranteed that the solution point will be the global minimum-distance point. 

Flatness in the case of (2.56) means that one seeks a small w [4], One way to ensure this is to minimize the norm, i-e. w = (w w). We can write this problem as a convex optimization problem ... 

Astrom, K. J., H. Panagopoulos, and T. Hagglund, Design of PI Controllers Based on Non-Convex Optimization, Automatica, 34, 585 (1998). 

A short list of such tools is reported in Table 1, where GUI and MOO indicate, respectively, if the software has a Graphical User Interface and if it allows for multi-objective optimization. We also indicate if each tool is open-source and if it allows for customization of the optimization algorithms. We should note that we included in the table only multi-disciplinary optimization tools that are specifically based on CIO methods, while we excluded software based on classic techniques for convex optimization, integer linear/non-linear programming and methods addressing combinatorial optimization only. We also excluded those technical software products that are not devoted specifically to optimization but still may include optimization methods, such as Matlab (which provides the Optimization Toolbox), and other CAD/CAE or multi-physics software, as well as multi-disciplinary tools that provide (as an extra feature) one or more, often domain-specific, optimization techniques (see for instance AVL CAMEO [23]). 

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