Solution Of Nonlinear Programming Problem Research Articles

Heap-Based Optimizer Algorithm with Chaotic Search for Nonlinear Programming Problem Global Solution

In this paper, a heap-based optimizer algorithm with chaotic search has been presented for the global solution of nonlinear programming problems. Heap-based optimizer (HBO) is a modern human social behavior-influenced algorithm that has been presented as an effective method to solve nonlinear programming problems. One of the difficulties that faces HBO is that it falls into locally optimal solutions and does not reach the global solution. To recompense the disadvantages of such modern algorithm, we integrate a heap-based optimizer with a chaotic search to reach the global optimization for nonlinear programming problems. The proposed algorithm displays the advantages of both modern techniques. The robustness of the proposed algorithm is inspected on a wide scale of different 42 problems including unimodal, multi-modal test problems, and CEC-C06 2019 benchmark problems. The comprehensive results have shown that the proposed algorithm effectively deals with nonlinear programming problems compared with 11 highly cited algorithms in addressing the tasks of optimization. As well as the rapid performance of the proposed algorithm in treating nonlinear programming problems has been proved as the proposed algorithm has taken less time to find the global solution.

Comparative Analysis of Nonlinear Programming Solvers: Performance Evaluation, Benchmarking, and Multi-UAV Optimal Path Planning

In this paper, we propose a set of guidelines to select a solver for the solution of nonlinear programming problems. We conduct a comparative analysis of the convergence performances of commonly used solvers for both unconstrained and constrained nonlinear programming problems. The comparison metrics involve accuracy, convergence rate, and computational time. MATLAB is chosen as the implementation platform due to its widespread adoption in academia and industry. Our study includes solvers which are either freely available or require a license, or are extensively documented in the literature. Moreover, we differentiate solvers if they allow the selection of different optimal search methods. We assess the performance of 24 algorithms on a set of 60 benchmark problems. We also evaluate the capability of each solver to tackle two large-scale UAV optimal path planning scenarios, specifically the 3D minimum time problem for UAV landing and the 3D minimum time problem for UAV formation flying. To enrich our analysis, we discuss the effects of each solver’s inner settings on accuracy, convergence rate, and computational time.

A New Approach for Kuhn-Tucker Conditions to Solve Quadratic Programming Problems with Linear Inequality Constraints

Many real-life problems, such as economic, industrial, engineering to mention but a few has been dealt with, using linear programming that assumes linearity in the objective function and constraint functions. It is noteworthy that there are many situations where the objective function and / or some or all of the constraints are non-linear functions. It is observed that many researchers have laboured so much at finding general solution approach to Non-linear programming problems but all to no avail. Of the prominent methods of solution of Non-linear programming problems: Karush- Kuhn-Tucker conditions method and Wolf modified simplex method. The Karush-Kuhn-Tucker theorem gives necessary and sufficient conditions for the existence of an optimal solution to non-linear programming problems, a finite-dimensional optimization problem where the variables have to fulfill some inequality constraints while Wolf in addition to Karush- Kuhn-Tucker conditions, modified the simplex method after changing quadratic linear function in the objective function to linear function. In this paper, an alternative method for Karush-Kuhn-Tucker conditional method is proposed. This method is simpler than the two methods considered to solve quadratic programming problems of maximizing quadratic objective function subject to a set of linear inequality constraints. This is established because of its computational efforts.

Kaibel column: Modeling, optimization, and conceptual design of multi-product dividing wall columns

In this work, we present the modeling, optimization, and conceptual design of a dividing wall column for the separation of four products, commonly referred to in the literature as a Kaibel column, by implementing three different formulations: an NLP, an MINLP, and a GDP formulation. For its solution, we propose a rigorous tray-by-tray model and compared it to results from a commercial software, followed by its reformulation to include a mixed-integer nonlinear programming and a general disjunctive programming formulation to respond to the conceptual design problem attached to these complex configurations. Considering the proposed rigorous model and the two formulations, the Kaibel column is solved, obtaining four high-purity products and new optimal tray locations for the feed and two side product streams, when the mixed-integer nonlinear programming formulation is applied. The use of these optimally located side streams showed reductions in the energy consumption when compared to cases were non-optimal fixed tray locations are used. When the general disjunctive programming problem was solved, the minimum number of trays needed in the main column and dividing wall are obtained, showing a great reduction of the remixing effects in the Kaibel column, and with that, a more energy efficient configuration. The models were coded in Pyomo using the solver IPOPT for the solution of the nonlinear programming problem, the solver Bonmin for the solution of the mixed-integer nonlinear programming problem, and GDPopt for the solution of the general disjunctive programming optimization problem.

On the stability of reinforced arches

This paper deals with the problems of stability of circular arches supported by inextensible threads that cannot withstand compressive forces. Both ends of the thread are attached to the axis of the arch so that the distance between the attachment points cannot increase during deformation. The problems of stability and supercritical behavior of elastic systems in the presence of unilateral restrictions on the movement lead to the need to study the bifurcation points of the equations or to find the parameters for which some variational problem with restrictions on the desired functions in the form of inequalities has a non-unique solution. In the numerical study, this problem is reduced to finding and studying the bifurcation points of solutions to a nonlinear programming problem. The problem of finding bifurcation points for solutions of nonlinear programming problems is reduced to the problem of identification of conditional positive definiteness of quadratic forms on cones. There are criteria of conditional positive definiteness of quadratic forms on cones in the important special case, when the cone is a positive orthant in Euclidean space. Their application poses the necessity to calculate a large number of determinants, which is computationally extremely inefficient, although in the case of a small number of variables, they offer promise for solving Lyapunov stability problems. The paper proposes and justifies the method of searching for options to solve the problem of nonconvex quadratic programming that arises during the study of the stability of elastic unilateral restrictions on the movement. The results can be used in the design of arches and arch systems to improve their bearing capacity.

IMPROVED PARTICLE SWARM OPTIMIZATION FOR NON-LINEAR PROGRAMMING PROBLEM WITH BARRIER METHOD

The Non-Linear Programming Problems (NLPP) are computationally hard to solve as compared to the Linear Programming Problems (LPP). To solve NLPP, the available methods are Lagrangian Multipliers, Sub gradient method, Karush-Kuhn-Tucker conditions, Penalty and Barrier method etc. In this paper, we are applying Barrier method to convert the NLPP with equality constraint to an NLPP without constraint. We use the improved version of famous Particle Swarm Optimization (PSO) method to obtain the solution of NLPP without constraint. SCILAB programming language is used to evaluate the solution on sample problems. The results of sample problems are compared on Improved PSO and general PSO.

FORMATION OF FUNCTIONAL VOLUME AND WORKING GROUPS OF ERP-PROJECT OF THE ENTERPRISE

Critical in the creation of ERP-system of the enterprise is the first stage of the system. Simultaneous introduction of all services for all business processes of the enterprise, as a rule, isn’t possible or expedient for various reasons: lack of sufficient resources, difficulties of creation of necessary technical infrastructure in short terms, big risks of radical reorganization of the operating control system, etc. Therefore, the functional volume of the 1st turn of the project joins the IT services providing effective functioning of the most interconnected business processes. The paper presents a mathematical formulation and procedure for solving the problem of determining the portfolio of IT services that realize the functions of the main business processes and make up the functional volume of the first stage of the ERP system of a large metallurgical company. Problem definition considers three important indicators of efficiency of the ERP-project: size of the project budget, number of communications between the IT services making the functional volume of the project (its “integration kernel”), number of the inherited design decisions from the systems of a “scrappy” computerization finding application in the enterprise resource planning. The first of these indicators is accepted as criterion of a task, other two – as its restrictions. As a result of formalization, the problem under consideration reduces to the problem of integer-valued quadratic programming. The method of network programming in modification used for the solution of nonlinear programming problems of is applied for its decision. The method decomposes a complex initial problem into a generalized dual task of network programming and a set of essentially simpler subtasks. An example is given of the solution of the problem under study in which the generalized dual problem of network programming is solved by the method of Lagrange multipliers, and individual subtasks are solved by the method of dichotomous programming, which reduces the solution of subproblems to the solution of a sequence of simple estimation problems. The solution of the task of forming a portfolio of services delivers the initial data for the task of scheduling the implementation of services by project groups responsible for the computerization of individual business processes of the enterprise.

Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM

In this paper, a class of Nonlinear Programming problem is modeled with gradient based system of fractional order differential equations in Caputo's sense. To see the overlap between the equilibrium point of the fractional order dynamic system and theoptimal solution of the NLP problem in a longer timespan the Multistage Variational ?teration Method isapplied. The comparisons among the multistage variational iteration method, the variationaliteration method and the fourth order Runge-Kutta method in fractional and integer order showthat fractional order model and techniques can be seen as an effective and reliable tool for finding optimal solutions of Nonlinear Programming problems.

Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

In this work, we show that, given a nonlinear programming problem, it is possible to construct a family of dynamical systems, defined on the feasible set of the given problem, so that: (a) the equilibrium points are the unknown critical points of the problem, which are asymptotically stable, (b) each dynamical system admits the objective function of the problem as a Lyapunov function, and (c) explicit formulas are available without involving the unknown critical points of the problem. The construction of the family of dynamical systems is based on the Control Lyapunov Function methodology, which is used in mathematical control theory for the construction of stabilizing feedback. The knowledge of a dynamical system with the previously mentioned properties allows the construction of algorithms, which guarantee the global convergence to the set of the critical points.

Characterization of optimal solutions for nonlinear programming problems with conic constraints

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ℝ n ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.

Optimal shakedown design of frames under stability conditions according to standards

A shakedown-frame plastic moments minimization and load-optimization nonlinear mathematical model with strength, stiffness, and stability constraints is investigated. A methodology and algorithms for stability evaluation have been developed according to various standards (Eurocode 3 (EC3) and the Dutch NEN 6771) by integrating the MatrixFrame commercial software for the building industry and the nonlinear mathematical programming software developed by the authors. For other investigators, this work makes it possible to integrate the solutions of nonlinear programming problems (plastic state variables - residual forces and displacements) into their structural design software. Numerical examples of optimization of frame structures are presented.

On an approach to the solution of nonlinear programming problems in inexact information on criterion importance coefficients and scale gradation in the additive value function

It is shown that the well-known approach to solving the problem of the minimization of the bilinear aim function with linear limitations on the criterion importance coefficients and values in terms of reducing it to linear programming via a special variable substitution is erroneous.

Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Since hyperbolic equations usually propagate discontinuities of initial or boundary conditions into the domain on which the solution lives or can develop discontinuities even in the presence of smooth data, problems of this type constitute a severe challenge for both theory and numerics of PDE constrained optimization. In the present paper, optimal control problems for the well-known wave equation are investigated. The intention is to study the order of the numerical approximations for both the optimal state and the optimal control variables for problems with known analytical solutions. The numerical method chosen here is a full discretization method based on appropriate finite differences by which the PDE constrained optimal control problem is transformed into a nonlinear programming problem (NLP). Hence, we follow here the approach ‘first discretize, then optimize’, which allows us to make use not only of powerful methods for the solution of NLPs, but also to compute sensitivity differentials, a necessary tool for real-time control.

Inner solvers for interior point methods for large scale nonlinear programming

This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative solvers, with an adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current outer iterate is far from the solution. In this work, we analyse the method of multipliers and the preconditioned conjugate gradient method as inner solvers for interior point schemes. We discuss the convergence of the whole approach, the implementation details and report the results of numerical experimentation on a set of large scale test problems arising from the discretization of elliptic control problems. A comparison with other interior point codes is also reported.

Optimal shapes of contact interfaces due to sliding wear in the steady relative motion

The transient wear process at contact frictional interface of two elastic bodies in relative steady motion induces evolution of shape of the interface. A steady wear state may be reached with uniform wear rate and fixed contact surface shape. In this paper, the optimal contact shape is studied by formulating several classes of shape optimization problems, namely minimization of generalized wear volume rate, friction dissipation power and wear dissipation rate occurring in two bodies. The wear rule was assumed as a nonlinear dependence of wear rate on friction traction and relative sliding velocity, similar to the Archard rule. The wear parameters of two bodies may be different. It was demonstrated that different optimal contact shapes are generated depending on objective functional and wear parameters. When the uniform wear rate is generated at contact sliding surfaces, the steady state is reached. It was shown that in the steady state the wear parameters of two bodies cannot be independent of each other. The solution of nonlinear programming problem was provided by the iterative numerical procedure. It was assumed that the relative sliding velocity between contacting bodies results from translation and rotation of two bodies. In general, both regular and singular regimes of wear rate and pressure distribution may occur. The illustrative examples of drum brake, translating punch and rotating annular punch (disc brake) provide the distribution of contact pressure and wear rate for regular and singular cases associated with the optimality conditions. It is shown that minimization of the generalized wear dissipation rate provides solutions assuring existence of steady wear states.

Modal Trimming Method for Improving Local Optima of Nonlinear Programming Problems

A method for improving local optimal solutions of nonlinear programming problems by treating constraints directly, which is named Modal Trimming Method, is proposed. This method combines a gradient method for searching local optimal solutions, with an extended Newton-Raphson method based on the Moore-Penrose generalized inverse of a Jacobian matrix of objective and constraint functions for searching initial feasible solutions used by the gradient method. A strategy for preventing traps into fathomed local optimal solutions is proposed. Some features of the method are investigated qualitatively, and it is found that they are suitable to improve local optimal solutions efficiently. The method is applied to a wide range of problems, and its performance is evaluated in terms of the global optimality of the suboptimal solutions. It turns out that the method has a high possibility of deriving the global optimal solutions by the suboptimal ones for a wide range of problems.

Convex Envelopes of Monomials of Odd Degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), particularly within the context of spatial Branch-and-Bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, x2k+1, where k ∈ N and the range of x includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.

1305 モードトリミング法による大域的最適化における不等式制約条件考慮の一方法

A method for improving local optimal solutions of nonlinear programming problems by treating equality constraints directly, named "Modal Trimming Method, " has been proposed, and it has turned out that the method has a high possibility of deriving the global optimal solutions by the suboptimal ones for a wide range of problems. It has been shown that this feature is because the renewal of solutions based on the extended Newton-Raphson method creates a chaotic behavior of the solutions, and a strategy for improving the capability of global search has been proposed. In this paper, a strategy for treating inequality constraints directly is proposed, and it turns out that the trap into local optimal solutions located on the boundaries of inequality constraints can be avoided. The modal trimming method with these strategies is applied to a variety of test problems, and its validity and effectiveness are clarified.

107 モードトリミング法におけるカオス発生条件の検討と大域的探索能力の向上

A method for improving local optimal solutions of nonlinear programming problems by treating constraints directly, named "Modal Trimming Method, " has been proposed, and it has turned out that the method has a high possibility of deriving the global optimal solutions by the suboptimal ones for a wide range of problems. It has also been shown that the aforementioned feature, i.e., the capability of the global search for deriving the global optimal solutions, is because the renewal of solutions based on the extended Newton-Raphson method creates a chaos behavior of the solutions. In this paper, the mechanism and conditions for the chaos behavior are clarified, and a strategy for improving the capability of global search is proposed.

204 非線形計画問題の局所的最適解向上のためのモードトリミング法 : 手法の提案と基本性能の評価

Stochastic methods have been proposed to derive global optimal solutions of nonlinear programming problems with any objective functions and constraints, and constraints have been treated indirectly by the penalty method. In this paper, a method of treating constraints directly is proposed to improve local optimal solutions of nonlinear programming problems, and is named modal trimming method. This method combines a gradient method for searching local optimal solutions, with an extended Newton-Raphson method based on the Moore-Penrose generalized inverse of a Jacobian matrix for searching initial feasible solutions used for the gradient method. To prevent traps into fathomed local optimal solutions, a strategy of transforming objective functions is adopted. Some features of the modal trimming method are investigated qualitatively, and the method is applied to small-scale optimization problems with strongly nonlinear objective functions and constraints to investigate its validity and effectiveness.