Unconstrained Nonlinear Programming Problem Research Articles

Matching Supply and Demand with Lead-Time Dependent Price and with Safety Stocks in a Make-to-Order Production System

We studied the ability to reduce the supply–demand mismatch of a periodic Make-to-Order (MTO) production system using safety stocks with marketing managing demand using lead-time guarantee and price as levers. The aim is to understand the interdependencies between lead-time guarantee, price, and safety stocks. We modeled the problem as an unconstrained stochastic non-linear programming problem, maximizing the expected profit per-unit time and obtaining a closed-form solution. The price is a function of the lead-time guarantee. Based on the sensitivity analysis of problem parameters, we found that lead-time competitiveness is adversely affected by a low safety stock level, MTO production rate (i.e., low supply capability), and product price (i.e., high demand volume). A shorter lead-time requires higher safety stock through reduced product and inventory holding costs. A higher price for a shorter lead-time in a lead-time-sensitive market reduces the safety stock. In a price-sensitive market, lead-time is decreased instead of the price. Demand variation results in longer lead-time and higher safety stock (provided the holding cost is low). For a higher price premium, price increases and lead-time decrease (safety stock increases). The integrated operation-marketing model captures the complex trade-offs not seen in a hierarchical model to produce better solutions.

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

Solution of boolean quadratic programming problems by two augmented Lagrangian algorithms based on a continuous relaxation

Many combinatorial optimization problems and engineering problems can be modeled as boolean quadratic programming (BQP) problems. In this paper, two augmented Lagrangian methods (ALM) are discussed for the solution of BQP problems based on a class of continuous functions. After convexification, the BQP is reformulated as an equivalent augmented Lagrangian function, and then solved by two ALM algorithms. Within this ALM algorithm, L-BFGS is called for the solution of unconstrained nonlinear programming problem. Experiments are performed on max-cut problem, 0–1 quadratic knapsack problem and image deconvolution, which indicate that ALM method is promising for solving large scale BQP by the quality of near optimal solution with low computational time.

Collaborative model for a two-echelon supply chain with uncertain demand under carbon tax policy

Consideration of carbon policies while optimizing supply chain operations has become imperative as governments and regulatory bodies throughout the world have implemented different carbon policies to reduce emissions of greenhouse gasses, especially CO2. “Carbon tax” is one of the major carbon policies, initiated by several governments to curtail emissions. In this study, we have considered this policy to optimize the total expected cost (TEC) of a two-echelon integrated supply chain with stochastic demand, where both backorders and lost sales are permitted. An unconstrained mixed integer non-linear programming (MINLP) problem is formulated and further solved to find the optimal production rate, order quantity, number of shipments and reorder point while minimizing the TEC. Emissions from all the major sources such as production, inventory and transportation have been taken into consideration. It is assumed that the emission from production is a function of production rate, and emission from transportation depends on payload and vehicle type. This study will help organizations to reduce cost and emissions, and regulatory bodies to decide proper tax rate on carbon emission.

Fast solution of minimum-time low-thrust transfer with eclipses

Optimizing low-thrust orbital transfers with eclipses by indirect methods raises several issues, namely the costate discontinuities at the eclipse entrance and exit, the initial costate guess sensitivity and the numerical accuracy required by the shooting method. The discontinuity issue is overcome by detecting the eclipse within the simulation and applying the costate jump derived analytically from the shadow constraint function. By fixing completely the targeted final position and velocity, the transversality conditions are removed and the shooting problem is recast as an unconstrained nonlinear programming problem. The numerical sensitivity issues are alleviated by using a derivative-free algorithm. The search space is reduced to four angles taking near zero values. This procedure yields a quasi-optimal solution from scratch in few minutes without requiring any specific user’s guess or tuning. The method is applicable whatever the thrust level and the eclipse configuration, as illustrated on transfers towards the geostationary orbit.

A Strut Finite Element for Exact Incompressible Isotropic Hyperelastic Analysis

Abstract This text describes a mathematical model of a strut finite element for isotropic incompressible hyperelastic materials. The invariants of the Right Cauchy-Green deformation tensor are written in terms of nodal displacements. The equilibrium problem is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy of the structure and the nodal displacements are the unknowns. The constraint for incompressibility is satisfied exactly, thereby eliminating the need for a penalty function. The results of the examples calculated by the proposed mathematical model show five significant digits in agreement when compared with commercial finite element analysis software.

Data-Based Predictive Control for Wastewater Treatment Process

Wastewater treatment process (WWTP) has long been a challenging industrial issue due to its built-in uncertainties and discontinuous measurement of system states. To solve this problem, in this paper, a data-based predictive control (DPC) strategy, based on the available sensing measurements, is proposed to control the dissolved oxygen (DO) concentration in WWTP. First, a self-organizing fuzzy neural network, which can adjust both the structure and parameters simultaneously, is developed to identify the real-time states of WWTP. Second, an improved nonlinear predictive control method is designed to reduce the online computation complexity by transforming the constrained conditions into an unconstrained nonlinear programming problem. Then, an adaptive second-order Levenberg-Marquardt algorithm is developed to derive the control law of DPC. Third, the theoretical analysis on the stability is also given to confirm the prerequisite of any successful application of DPC. Finally, the proposed DPC strategy is applied to the Benchmark Simulation Model No. 1. Experimental results demonstrate that the control performance of the proposed DPC is better than some existing methods.

Phase-Only Zero Forcing for Secure Communication With Multiple Antennas

Artificial noise (AN) transmission can enhance the secrecy in multiantenna wireless channels by superimposing judiciously constructed synthetic noise signals over information signals during transmission. Motivated by the large-scale multiple-input multiple-output systems, we study AN transmission under a constraint that all elements (weights) in each beamforming vector must have a constant magnitude, but can have arbitrary phases. For the special case of one AN beamforming vector and one legitimate receiver, we derive a necessary and sufficient condition for finding a beamforming vector in the null space of the legitimate receiver's channel and provide a geometric interpretation. For the independent identically distributed Rayleigh fading channel, we derive an approximate expression for the probability of failing to find such a beamforming vector, and show that it decreases exponentially in the square of the number of transmit antennas. For the general case, we propose a numerical algorithm for obtaining a set of mutually orthogonal AN beamforming vectors in the null space of all the receivers. Our approach involves reducing the problem to an unconstrained nonlinear programming problem, which is then solved using the Gauss–Newton method. We show numerically that our proposed algorithm performs significantly better than a heuristic relaxation approach. Finally, for a multiantenna system with multiple RF chains, we show numerically that the secrecy rate achieved by our proposed approach is close to that achieved by the AN transmission with unconstrained beamforming, when the number of transmit antennas is sufficiently large.

Solving volterra integral equation via nonlinear programming

In this paper we propose an approach to find approximate solution to the nonlinear Volterra integral equation of the second type through a nonlinear programming technique by firstly converting the integral equation into a least square cost function as an objective function for an unconstrained nonlinear programming problem which solved by a nonlinear programming technique (The preconditioned limited- memory quasi-Newton conjugates, gradient method) and as far as we read this is a new approach in the ways of solving the nonlinear Volterra integral equation. We use Maple 11 software as a tool for performing the suggested steps in solving the examples.

Passive Mechanical Control With a Special Class of Positive Real Controllers: Application to Passive Vehicle Suspensions

This paper presents an efficient H2 optimization method for passive mechanical control problems with a special class of positive real controllers. In particular, the problem of designing passive vehicle suspensions based on a full-car model is taken as an example, where both the positive real constraint and the constraint imposed on the static stiffness are considered. An unconstrained nonlinear programming problem is formulated by using the structured H2 optimization framework, and the Lagrange matrix multiplier method is employed to derive a set of necessary conditions for the optimization so that the time-efficient gradient-based algorithms can easily be implemented. The proposed method can also effectively deal with the fixed static stiffness optimization problem and it is shown in the numerical examples that the proposed method cannot only recover the existing fixed-structure configuration, but also introduce new (optimal) configurations with respect to the specific weighting factors, which demonstrates the effectiveness of the proposed method.

Observer-based reliable stabilization of uncertain linear systems subject to actuator faults, saturation, and bounded system disturbances

A matrix inequality approach is proposed to reliably stabilize a class of uncertain linear systems subject to actuator faults, saturation, and bounded system disturbances. The system states are assumed immeasurable, and a classical observer is incorporated for observation to enable state-based feedback control. Both the stability and stabilization of the closed-loop system are discussed and the closed-loop domain of attraction is estimated by an ellipsoidal invariant set. The resultant stabilization conditions in the form of matrix inequalities enable simultaneous optimization of both the observer gain and the feedback controller gain, which is realized by converting the non-convex optimization problem to an unconstrained nonlinear programming problem. The effectiveness of proposed design techniques is demonstrated through a linearized model of F-18 HARV around an operating point.

A Generalized Computer Technique for Solving Unconstrained Non-Linear Programming Problems

An unconstrained problem with nonlinear objective function has many applications. This is often viewed as a discipline in and of itself. In this paper, we develop a computer technique for solving nonlinear unconstrained problems in a single framework incorporating with Golden section, Gradient Search method. For this, we first combine this algorithm and then develop a generalized computer technique using the programming language MATHEMATICA. We demonstrate our computer technique with a number of numerical examples. Dhaka Univ. J. Sci. 61(1): 75-80, 2013 (January) DOI: http://dx.doi.org/10.3329/dujs.v61i1.15100

Best proximity point theorems: exposition of a significant non-linear programming problem

The primary goal of this work is to address the non-linear programming problem of globally minimizing the real valued function x ? d(x, Tx) where T is presumed to be a non-self mapping that is a generalized proximal contraction in the setting of a metric space. Indeed, an iterative algorithm is presented to determine a solution of the preceding non-linear programming problem that focuses on global optimization. As a sequel, one can compute optimal approximate solutions to some fixed point equations and optimal solutions to some unconstrained non-linear programming problems.

Scale Efficiency Measurement in Data Envelopment Analysis with Interval Data: A Two-Level Programming Approach

Conventional data envelopment analysis (DEA) for measuring the relative efficiency of a set of decisionmaking units (DMUs) requires the observations to have precise values. When observations are imprecise and represented by interval values, the efficiencies are also expected to reflect interval values. Several methods exist to calculate the interval overall and technical efficiencies, but such methods are unable to calculate the interval scale efficiency. The focus of this paper is the application of a two-level programming technique to formulate the problem of determining the bounds of the interval scale efficiency. The associated models are essentially nonlinear programs with only bound constraints for variables in a sophisticated form. Hence, one can modify the conventional quasi-Newton method for unconstrained nonlinear programming problems to solve the two-level programs. Two examples with interval data, one hypothetical and one real, aid in explaining the proposed method and the properties of the results.

A dynamic convexized method for nonconvex mixed integer nonlinear programming

We consider in this paper the nonconvex mixed-integer nonlinear programming problem. We present a mixed local search method to find a local minimizer of an unconstrained nonconvex mixed-integer nonlinear programming problem. Then an auxiliary function which has the same global minimizers and the same global minimal value as the original problem is constructed. Minimization of the auxiliary function using our local search method can escape successfully from previously converged local minimizers by taking increasing values of parameters. For the constrained nonconvex mixed-integer nonlinear programming problem, we develop a penalty based method to convert the problem into an unconstrained one, and then use the above method to solve the later problem. Numerical experiments and comparisons on a set of MINLP benchmark problems show the effectiveness of the proposed algorithm.

Nonlinear Predictive Control of Fed‐Batch Cultures of Escherichia coli

AbstractA strategy for controlling a fed‐batch Escherichia coli culture is described to maintain the culture at the boundary between oxidative and oxido‐fermentative regimes. A nonlinear predictive controller is designed to regulate the acetate concentration, constraining the feed rate to follow an optimal reference profile which maximizes the biomass growth. For the sake of simplicity and efficiency, the original problem is converted into an unconstrained nonlinear programming problem, solved by control vector parameterization techniques. The robustness of the structure is further improved by explicitly including the difference between system and model prediction. A robustness study based on a Monte Carlo approach is used to evaluate the performance of the proposed controller. This control law is finally compared to the generic model control strategy.

A new class of projection and contraction methods for solving variational inequality problems

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods.

Finite Word Length Implementation for Digital Reduced Order Observer Based Controllers

AbstractImplementation issues for digital reduced‐order observer‐based controllers with Finite Word Length (FWL) considerations is studied. A tractable FWL stability related measure is derived, and the optimal FWL realization problem for digital reduced‐order obverser‐based controller is to find those realizations that maximize this related measure. This optimization problem is formulated as an unconstrained nonlinear programming problem which can be solved using the simplex search algorithm. A numerical example is given to illustrate the design procedure and the effectiveness of the proposed method.

Solving Variational Inequalities in a Fuzzy Environment

This paper deals with the variational inequalities in a fuzzy environment. It shows that solving the fuzzy variational inequalities is equivalent to solving a fuzzy generalized complementarity problem. By using the tolerance approach, we show that the fuzzy generalized complementarity problem can be converted to a regular unconstrained nonlinear programming problem. An entropic regularization technique is then applied to solve the resulting optimization problem. Some computational results are included.

Optimal development of doubly curved surfaces

This paper presents algorithms for optimal development (flattening) of a smooth continuous curved surface embedded in three-dimensional space into a planar shape. The development process is modeled by in-plane strain (stretching) from the curved surface to its planar development. The distribution of the appropriate minimum strain field is obtained by solving a constrained nonlinear programming problem. Based on the strain distribution and the coefficients of the first fundamental form of the curved surface, another unconstrained nonlinear programming problem is solved to obtain the optimal developed planar shape. The convergence and complexity properties of our algorithms are analyzed theoretically and numerically. Examples show the effectiveness of the algorithms.