Optimization quadratic programming

Keywords, protein folding, tertiary structure, potential energy surface, global optimization, empirical potential, residue potential, surface potential, parameter estimation, density estimation, cluster analysis, quadratic programming... 

Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equahty and inequahty constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generaUy favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

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)... 

Problem 4.1 is nonlinear if one or more of the functions/, gv...,gm are nonlinear. It is unconstrained if there are no constraint functions g, and no bounds on the jc,., and it is bound-constrained if only the xt are bounded. In linearly constrained problems all constraint functions g, are linear, and the objective/is nonlinear. There are special NLP algorithms and software for unconstrained and bound-constrained problems, and we describe these in Chapters 6 and 8. Methods and software for solving constrained NLPs use many ideas from the unconstrained case. Most modem software can handle nonlinear constraints, and is especially efficient on linearly constrained problems. A linearly constrained problem with a quadratic objective is called a quadratic program (QP). Special methods exist for solving QPs, and these iare often faster than general purpose optimization procedures. 

Fan, Y. S. Sarkar and L. Lasdon. Experiments with Successive Quadratic Programming Algorithms. J Optim Theory Appli 56 (3), 359-383 (March 1988). 

Schmid, C. and L. T. Biegler. Reduced Hessian Successive Quadratic Programming for Real Time Optimization. Proceed IFAC Adv Control Chem Processes, Kyoto, Japan, 173-178 (1994). 

Sequential quadratic programming. A sequential quadratic programming (SQP) technique involves the resolution of a sequence of explicit quadratic programming (QP) subproblems. The solution of each subproblem produces the search direction d that has to be taken to reach the next iterate zk+i from the current iterate zk. A one-dimensional search is then accomplished in the direction dt to obtain the optimal step size. 

The above problem becomes an NLP problem when we fix the integer variables and since we have only 10 feasible compounds, 10 NLP problems were solved by fixing the binary variables representing the 10 compounds. Sequential quadratic programming algorithm was used to solve the NLP problems. The molecular structure and design results of the optimal solvent and 2-ethoxy ethyl acetate are shown in Table 1. 

For process optimization problems, the sparse approach has been further developed in studies by Kumar and Lucia (1987), Lucia and Kumar (1988), and Lucia and Xu (1990). Here they formulated a large-scale approach that incorporates indefinite quasi-Newton updates and can be tailored to specific process optimization problems. In the last study they also develop a sparse quadratic programming approach based on indefinite matrix factorizations due to Bunch and Parlett (1971). Also, a trust region strategy is substituted for the line search step mentioned above. This approach was successfully applied to the optimization of several complex distillation column models with up to 200 variables. 

Biegler, L. T., Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Comp, and Chem. Eng. 8(3/4), 243-248 (1984). 

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. 

Wright, S. E., Foley, F. A., and Flughes, J. M. (2000). Optimization of site occupancies in minerals using quadratic programming. Amer. Miner. 85, 524-31. 

P. M. Pardalos and G. Schnitger. Checking local optimality in constrained quadratic programming is NP-Hard. Oper. Res. Lett., 7(1) 33,1988. 

P. M. Pardalos and S. A. Vavasis. Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim., 1 15,1991. 

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