Modified limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for unconstrained optimization problem

The use of the self-scaling Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is very efficient for the resolution of large-scale optimization problems, in this paper, we present a new algorithm and modified the self-scaling BFGS algorithm. Also, based on noticeable non-monotone line search properties, we discovered and employed a new non-monotone idea. Thereafter first, an updated formula is exhorted to the convergent Hessian matrix and we have achieved the secant condition, second, we established the global convergence properties of the algorithm under some mild conditions and the objective function is not convexity hypothesis. A promising behavior is achieved and the numerical results are also reported of the new algorithm.

Data-Driven Self-Optimizing Control: Unconstrained Optimization Problem

Controlled variable (CV) selection plays an important role in determining the performance of a process plant. Existing methods for CV selection through self-optimizing control requires linearization of rigorous models around nominal operating points. This is a very difficult task which results to large losses. This work presents a novel method for CV design. A necessary condition of optimality (NCO) was proposed to be the CV. The approach does not require the analytical expression of the NCO to be derived but is approximated through a single regression step based on data. Finite difference was used to approximate the NCO (gradient) using data; three finite difference schemes were employed for this purpose, which are forward, backward and central differences. Seven different cases with respect to number of sampling points, neighborhood points and finite difference schemes were investigated. To demonstrate the efficacy of the method in simplest way, it is applied to a hypothetical unconstrained optimization problem. The proposed method was found to have outperformed some existing approaches in many instances. A zero loss was recorded by some designed CVs. Central difference was found to be the best schemes among the three. Keywords— controlled variable, disturbance, finite difference, monotonicity, regression.

Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

A modified filter nonmonotone adaptive retrospective trust region method

In this paper, aiming at the unconstrained optimization problem, a new nonmonotone adaptive retrospective trust region line search method is presented, which takes advantages of multidimensional filter technique to increase the acceptance probability of the trial step. The new nonmonotone trust region ratio is presented, which based on the convex combination of nonmonotone trust region ratio and retrospective ratio. The global convergence and the superlinear convergence of the algorithm are shown in the right circumstances. Comparative numerical experiments show the better effective and robustness.

A Functional Interpolation Approach to Compute Periodic Orbits in the Circular-Restricted Three-Body Problem

In this paper, we develop a method to solve for periodic orbits, i.e., Lyapunov and Halo orbits, using a functional interpolation scheme called the Theory of Functional Connections (TFC). Using this technique, a periodic constraint is analytically embedded into the TFC constrained expression. By doing this, the system of differential equations governing the three-body problem is transformed into an unconstrained optimization problem where simple numerical schemes can be used to find a solution, e.g., nonlinear least-squares is used. This allows for a simpler numerical implementation with comparable accuracy and speed to the traditional differential corrector method.

An Efficient Modified AZPRP Conjugate Gradient Method for Large-Scale Unconstrained Optimization Problem

To find a solution of unconstrained optimization problems, we normally use a conjugate gradient (CG) method since it does not cost memory or storage of second derivative like Newton’s method or Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. Recently, a new modification of Polak and Ribiere method was proposed with new restart condition to give a so-call AZPRP method. In this paper, we propose a new modification of AZPRP CG method to solve large-scale unconstrained optimization problems based on a modification of restart condition. The new parameter satisfies the descent property and the global convergence analysis with the strong Wolfe-Powell line search. The numerical results prove that the new CG method is strongly aggressive compared with CG_Descent method. The comparisons are made under a set of more than 140 standard functions from the CUTEst library. The comparison includes number of iterations and CPU time.

Optimization of the power consumption mode of pumping stations of “Suv Okova” by reactive power

The optimal modes of the existing compensating devices under operating conditions were determined. Minimum power and energy losses were taken as optimality criteria. The unconstrained optimization problem was considered, i.e., finding the absolute minimum. When solving the problem to find the optimal mode of operation of the “SUV OKOVA” network, a relative minimum was found since the system of constraints has a nonlinear form. The problem of conditional optimization in terms of reactive power is solved, for which the gradient method of quadratic programming is used.

Evaluation of the efficiency of the problem of choosing the extreme values of the parameters of the parametric family of regenerating processes

Thus, the formal formulation of the problem of evaluating the effectiveness of search optimization procedures on simulation models of regenerating processes under strict time constraints has been completed. A procedure for parametric tuning of the optimization algorithm has been developed, which sequentially refines the values of the functional specified by the model and redistributes the remaining model regeneration cycles between the investigated values of the controlled parameter. The problem of maximizing the probability of the correct choice is posed and solved, i.e. selection, according to the results of the simulation experiment on the model of the regenerating process, of the value of the controlled parameter that delivers the true maximum to the investigated functional. Based on the transition to the Lagrangian, the solution to the constrained optimization problem is reduced to an unconstrained optimization problem. Analytical expressions are obtained to assess the optimal distribution of regeneration cycles. It is shown that the simulation model with the included search engine optimization algorithm provides solutions that are quite effective in terms of computational costs. As a result, a method is proposed for a simple extension of the developed simulation models by including a search optimization algorithm, which makes it possible to move from modeling the system to optimizing its objective function on a given area of controlled parameters.

Spectral proximal method for solving large scale sparse optimization

In this paper, we propose to use spectral proximal method to solve sparse optimization problems. Sparse optimization refers to an optimization problem involving the ι0 -norm in objective or constraints. The previous research showed that the spectral gradient method is outperformed the other standard unconstrained optimization methods. This is due to spectral gradient method replaced the full rank matrix by a diagonal matrix and the memory decreased from Ο(n2) to Ο(n). Since ι0-norm term is nonconvex and non-smooth, it cannot be solved by standard optimization algorithm. We will solve the ι0 -norm problem with an underdetermined system as its constraint will be considered. Using Lagrange method, this problem is transformed into an unconstrained optimization problem. A new method called spectral proximal method is proposed, which is a combination of proximal method and spectral gradient method. The spectral proximal method is then applied to the ι0-norm unconstrained optimization problem. The programming code will be written in Python to compare the efficiency of the proposed method with some existing methods. The benchmarks of the comparison are based on number of iterations, number of functions call and the computational time. Theoretically, the proposed method requires less storage and less computational time.