Sequence Of Quadratic Programming Problems Research Articles

An efficient and provable sequential quadratic programming method for American and swing option pricing

A sequential quadratic programming numerical method is proposed for American option pricing based on the variational inequality formulation. The variational inequality is discretized using the θ-method in time and the finite element method in space. The resulting system of algebraic inequalities at each time step is solved through a sequence of box-constrained quadratic programming problems, with the latter being solved by a globally and quadratically convergent, large-scale suitable reflective Newton method. It is proved that the sequence of quadratic programming problems converges with a constant rate under a mild condition on the time step size. The method is general in solving the variational inequalities for the option pricing with many styles of optimal stopping and complex underlying asset models. In particular, swing options and stochastic volatility and jump diffusion models are studied. Numerical examples are presented to confirm the effectiveness of the method.

An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials

In this paper, a method for solving a class of nonlinear optimal control problems is presented. The method is based on replacing the dynamic nonlinear optimal control problem by a sequence of quadratic programming problems. To this end, the iterative technique developed by Banks is used to replace the original nonlinear dynamic system by a sequence of linear time-varying dynamic systems, then each of the new problems is converted to quadratic programming problem by parameterizing the state variables by a finite length Chebyshev series with unknown parameters. To show the effectiveness of the proposed method, simulation results of a nonlinear optimal control problem are presented.

Closed Form Solution of Nonlinear-Quadratic Optimal Control Problem by State-control Parameterization using Chebyshev Polynomials

In this paper the quasilinearization technique along with the Chebyshev polynomials of the first type are used to solve the nonlinear-quadratic optimal control problem with terminal state constraints. The quasilinearization is used to convert the nonlinear quadratic optimal control problem into sequence of linear quadratic optimal control problems. Then by approximating the state and control variables using Chebyshev polynomials, the optimal control problem can be approximated by a sequence of quadratic programming problems. The paper presents a closed form solution that relates the parameters of each of the quadratic programming problems to the original problem parameters. To illustrate the numerical behavior of the proposed method, the solution of the Van der Pol oscillator problem with and without terminal state constraints is presented.

A New Approach for Solving Optimal Nonlinear Control Problems Using Decriminalization and Rationalized Haar Functions

This paper presents a numerical method based on quasilinearization and rationalized Haar functions for solving nonlinear optimal control problems including terminal state constraints, state and control inequality constraints. The optimal control problem is converted into a sequence of quadratic programming problems. The rationalized Haar functions with unknown coefficients are used to approximate the control variables and the derivative of the state variables. By adding artificial controls, the number of state and control variables is equal. Then the quasilinearization method is used to change the nonlinear optimal control problems with a sequence of constrained linear-quadratic optimal control problems. To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

Combination of non-classical pseudospectral and direct methods for the solution of brachistochrone problem

In this paper, we propose a combination of non-classical pseudospectral and direct methods to find the solution of brachistochrone problem. The method converts the optimal control problem of brachistochrone, into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem into a sequence of constrained linear-quadratic optimal control problems; then each of the state variables is approximated by a weighted interpolation function based on the non-classical orthogonal polynomials. The method gives the information of the quadratic programming problems explicitly (the Hessian and the gradient of the cost function). Using this method, the solution of the brachistochrone problem is compared with those in the literature.

Solution of interface problems with nonmonotone contact and friction laws using a neural network optimization environment

The paper addresses the solution of interface problems which present a degrading behaviour leading to nonmonotone contact and friction laws. The paper is based on a superpotential formulation that leads to hemivariational inequalities, or, equivalently to nonconvex–nonsmooth optimization problems. The above problems are addressed with a heuristic method which effectively replaces the nonconvex–nonsmooth optimization problem by a sequence of quadratic programming problems with inequality constraints. Then, the quadratic optimization problems are transformed into a system of differential equations, which are treated numerically with an appropriate neural network. Finally, a numerical example illustrates the properties of the proposed algorithmic scheme.

Optimization algorithms for bilinear model–based predictive control problems

AbstractModel‐based predictive control (MPC) for discrete‐time bilinear state–space models is considered. The optimization problem of the bilinear MPC algorithm is nonlinear in general. It is demonstrated that the structural properties of the bilinear state–space model provide a way to formulate the nonlinear optimization problem as a sequence of quadratic programming problems that exactly represent the original objective function. The proposed optimization algorithm is compared to one that is based on a linearization about an input trajectory. To benefit from the advantages of both algorithms, a hybrid algorithm is proposed, which outperforms the other two in most cases. The applicability of the proposed bilinear MPC algorithm is demonstrated on a polymerization process. 2004 American Institute of Chemical Engineers AIChE J, 50:1453–1461, 2004

Successive approximation method for non‐linear optimal control problems with application to a container crane problem

AbstractA numerical method is proposed to solve a non‐linear optimal control problem with terminal state constraints, inequality state constraints and control saturation constraints. The method solves this problem directly and without employing penalty functions. First, the problem is reformulated, using the quasilinearization and the state parameterization techniques, into a sequence of quadratic programming problems. Then the resulting quadratic programming problems are solved by the active set method. The proposed method is applied to a practical optimal control problem of minimizing the swing angle of a container during its transfer from a ship to a truck. Copyright © 2002 John Wiley & Sons, Ltd.

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.

Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints

The present paper deals with the problem of determining the optimal joint positions and cross-sectional parameters of linearly elastic space frames with imposed stress and free frequency constraints. The frame is assumed to be acted on by different load systems, including temperature and self-weight loads. The stress state analysis includes tension, bending, shear, and torsion of beam elements. By a sequence of quadratic programming problems, the optimal design is attained. The sensitivity analysis of distinct as well as multiple frequencies is performed through analytic differentiation with respect to design parameters. Illustrative examples of optimal design of simple and medium complexity frames are presented, and the particular case of bimodal optimal solution is considered in detail.

Efficient algorithms for function approximation with piecewise linear sigmoidal networks

This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the well-known method of fitting the residual. The task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of quadratic programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g., backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, solutions that are independent of initial conditions, good scaling properties and a robust numerical implementation. Empirical results are included to illustrate these characteristics.

On design of joint positions for minimum mass 3D frames

An iterative procedure for determining the joint positions and dimensions of cross-sections corresponding to a minimum mass space frame is presented. Stress constraints, displacement constraints and side constraints are taken into account, with the possibility for linking of the design variables, which in addition to the joint positions are the beam dimensions. The frame is assumed to carry consecutively a number of different systems of loads, including temperature loads and weights. The stressed state analysis includes tension, bending, shear and torsion of the beam elements. The optimization iteration includes a sequence of quadratic programming problems with the possibility of move-limits. Only “nearby” constraints (active strategy) are considered, and analytical expressions for the gradients are derived. Illustrative medium scale problems are presented.

Numerical treatment of the sea bed-long underwater cable dynamic interaction problem by an upper bounding algorithm

The numerical treatment of the dynamic response of an underwater cable, constrained by a frictionless rigid sea profile, by the use of an upper bounding algorithm is considered. By applying an appropriate discretization scheme with respect to time and space, the arising unilateral contact/impact problem is formulated as a sequence of variational inequality problems or as a sequence of quadratic programming problems that are equivalent to linear complementarity problems. The latter are numerically treated by applying an easy-to-use upper bounding Mangasarian algorithm leading to the solution of a linear programming problem which permits the treatment of large-scale problems.

AnL 1 smoothing spline algorithm with cross validation

We propose an algorithm for the computation ofL1 (LAD) smoothing splines in the spacesWM(D), with\([0, 1]^n \subseteq D\). We assume one is given data of the formyi=(f(ti) +ei, i=1,...,N with {itti}i=1N ⊂D, theei are errors withE(ei)=0, andf is assumed to be inWM. The LAD smoothing spline, for fixed smoothing parameterλ⩾0, is defined as the solution,sλ, of the optimization problem\(\min _{g \in W_M }\) (1/N)∑i=1N¦yi−g(ti¦+λJM(g), whereJM(g) is the seminorm consisting of the sum of the squaredL2 norms of theMth partial derivatives ofg. Such an LAD smoothing spline,sλ, would be expected to give robust smoothed estimates off in situations where theei are from a distribution with heavy tails. The solution to such a problem is a “thin plate spline” of known form. An algorithm for computingsλ is given which is based on considering a sequence of quadratic programming problems whose structure is guided by the optimality conditions for the above convex minimization problem, and which are solved readily, if a good initial point is available. The “data driven” selection of the smoothing parameter is achieved by minimizing aCV(λ) score of the form\((1/N)[\sum\nolimits_{i = 1}^N {\left| {y_i - s_\lambda (t_i )} \right| + \sum\nolimits_{res_i = 0} 1 } ]\).The combined LAD-CV smoothing spline algorithm is a continuation scheme in λ↘0 taken on the above SQPs parametrized inλ, with the optimal smoothing parameter taken to be that value ofλ at which theCV(λ) score first begins to increase. The feasibility of constructing the LAD-CV smoothing spline is illustrated by an application to a problem in environment data interpretation.

Stochastic Technology and Prices: Comment

In a recent article Paris and Easter (PE) propose a nonlinear mathematical programming structure and solution procedure for examining production with stochastic prices and technology. Stochastic prices are handled with a mean-variance-type objective function. Stochastic technology is handled with chance constraints. The proposed solution procedure involves solving a sequence of quadratic programming problems that they claim will converge in the limit to the optimal solution. Based on this claim, they offer the procedure as a means by which to obtain close approximations of the optimal solution. This comment demonstrates that the proposed solution method cannot converge in the limit to the optimal solution except in special cases of limited interest. The implication is that the procedure cannot be relied on generally by researchers to obtain close approximations. The demonstration follows a brief review. The allocation problem PE propose and seek to solve is written:

Sea bed-structure interaction for submarine cables

In this study, the dynamic behaviour of a submarine cable constrained by a frictionless rigid sea profile is studied. After a suitable local and time discretization, the arising unilateral contact-impact problem is formulated as a sequence of variational inequality problems which lead equivalently to a sequence of quadratic programming problems. The solution method is based on some propositions which constitute the mechanical “interpretation” of some theorems of quadratic programming and combines the advantages of the optimization algorithms with the advantages of the “trial and error” methods. Finally the applicability of the proposed method is illustrated by means of a numerical example.

AN APPROACH TO STRUCTURAL OPTIMIZATION—SEQUENTIAL QUADRATIC PROGRAMMING, SQP

On the basis of previous works, here we propose an approach to the optimal design of engineering structures for member sizing and shape optimization. This approach is based on the transformation of a non-linear programming problem (NLP) into a sequence of quadratic programming problems (SQP). Compared with the commonly used sequential linear programming (SLP), SQP is more suited to the highly non-linear problems of structural optimization. Different modes of SQP may be used. Four types ofSQP presented in this paper have been demonstrated to be stable and rapid in their convergence according to our experiences. Quadratic programming (QP) has several reliable and efficient algorithms for its solution which require more computation than does linear programming, but owing to the reduced number of iterations in optimization, the total work involved in SQP is in general less than that of SLP. Further developments of SQP will be very helpful for structural optimization.