Nonlinear Programming Techniques Research Articles

Continuous Nonlinear Programming Techniques to Solve Scheduling Problems

One of the most known applications of Discrete Optimization is on scheduling. In contrast, one of the most known applications of Continuous Nonlinear Optimization is on the control of dynamic systems. In this paper, we combine both views, solving scheduling problems as dynamic systems, modeled as discrete-time nonlinear optimal control problems with state and control continuous variables subjected to upper and lower bounds. Complementarity constraints are used to represent scheduling decisions. One example we discuss in detail is the crude oil scheduling in ports, with numerical results presented.

Finite Element Limit Analysis of Passive Earth Resistance in Cohesionless Soils

This note examines the classic passive earth resistance of cohesionless soil by using two newly developed numerical procedures based on finite element formulations of the bound theorems of limit analysis and non-linear programming techniques. Solutions using upper and lower bounds are presented to complement the previous studies of this problem. The parameters studied are soil-wall interface friction, wall inclination, backfill surface configuration and the wall's weight.

Development of an optimal trajectory model for spray painting on a free surface

Generally, trajectory planning problems are often formulated as constrained variational problems. This paper develops an optimal trajectory model to optimize the trajectory planning on a free surface to achieve uniform deposition over painted surface and reduce wastage of coating materials. Numerical solution techniques consider geometric characteristics on a free surface, rate of film accumulation, and position and orientation of spray guns in uniformity coating and painting duration. Given a specified spatial path and functions of film accumulation rates for an infinite range model and a beta distribution model, some results are obtained by using nonlinear programming techniques based difference quasi-Newton method over cone surfaces. It provides objective functions that can be used to minimize the thickness variation of the paint spraying of a specified spatial path. These results also give an evaluation along the average velocity trajectory and the optimal trajectory to demonstrate the feasibility of the proposed methods. The formulated optimization methods can be applied to solve the same class of nonlinear planning problems.

Portfolio optimization with CVaR under VG process

Formal portfolio optimization methodologies describe the dynamics of financial instruments price with Gaussian Copula (GC). Without considering the skewness and kurtosis of assets return rate, optimization with GC underestimate the optimal CVaR of portfolio. In the present paper, we develop the approach for portfolio optimization by introducing Lévy processes. It focuses on describing the dynamics of assets’ log price with Variance Gamma copula (VGC) rather than GC. A case study for three Indexes of Chinese Stock Market is performed. On application purpose, we calculate the best hedge positions of Shanghai Index (SHI), Shenzhen Index (SZI) and Small Cap Index (SCI) with the performance function CVaR under VG model. It can be combined with Monte Carlo Simulation and nonlinear programming techniques. This framework is suitable for any investment companies.

A Systematic Design Flow for Optimal Capacitance Assignment in Switched-Capacitor Biquads

This paper presents a systematic design flow for optimal capacitance assignment in switched-capacitor biquad circuits. The proposed design flow starts with a biquad template configuration, which is defined based on circuit regularity. Then, the symbolic transfer function in the z-domain is derived for the biquad template. Given the desired transfer function to be realized, a set of equality constraints can be obtained from coefficient matching of the symbolic transfer function and the desired one. In addition, we also formulate an exact closed-form expression for the output swing constraint so that output swings for the biquad are equalized. Other constraints, such as direct charge coupling, are also formulated. Finally, a nonlinear programming technique is used to solve for capacitance variables subject to optimal capacitance assignment, which is defined as minimizing total capacitance, total number of capacitors and circuit sensitivity. The main advantage of the proposed design flow is that an optimal biquad configuration can be obtained in an automated and accurate way. Experiments confirmed the flexibility of the design flow on obtaining optimal biquad configurations under various design tradeoffs.

An equilibrium analysis of the revenue-maximizing multinational enterprise

Departing from the traditional profit-maximization assumption underlying the theory of the multinational enterprises (MNEs), this paper proposes a model of the revenue-maximizing MNE subject to the profit constraint. The nonlinear programming techniques are used for the equilibrium analysis of the MNE’ s decision-making. A set of optimal conditions is derived regarding the quantity of output, volume of intra-firm trade, allocation of multinational production, as well as transfer prices in both exogenous and endogenous cases. Under the context, a comparison of the behavioral differences is made between the revenue-maximizing MNE and the profit-maximizing MNE.

A modified hybrid differential evolution for short-term scheduling of hydrothermal power systems with cascaded reservoirs

This paper presents a new approach for short-term hydrothermal scheduling of cascaded reservoirs using a modified hybrid differential evolution (MHDE) algorithm. Hydrothermal scheduling involves optimization of a non-linear objective function with a set of system and hydraulic constraints. In this paper the hybrid differential algorithm is modified in order to handle the equality constraints especially reservoir end volume and power balance constraints. The hybrid differential evolution in combination with a novel equality constraint handling mechanism provides better solution at a lesser computational effort. To show the efficiency of the proposed algorithm different case studies are carried out considering valve-point effects, prohibited discharge zones, multiple thermal units and transmission losses. The results of the proposed approach are compared with those of dynamic programming, non-linear programming, genetic algorithm, evolutionary programming and particle swarm optimization techniques. The results of the proposed algorithm are compared with differential evolution (DE) and its variants. From the numerical results it is found that the MHDE based approach is able to provide better solution at a relatively lesser computational effort.

Using Non-Linear Programming Techniques in Determination of the Most Probable Slip Surface in 3D Slopes

Using Non-Linear Programming Techniques in Determination of the Most Probable Slip Surface in 3D Slopes

Peak Load Management in Electrolytic Process Industries

Electrolytic process, employed for manufacturing basic chemicals like caustic soda and chlorine, is highly energy intensive. Due to escalating costs of fossil fuels and capacity addition, the electricity cost has been increasing for the last few decades. Electricity intensive industries find it very difficult to cope up with higher electricity charges particularly with time-of-use (TOU) tariffs implemented by the utilities with the objective of flattening the load curve. Load management programs focusing on reduced electricity use at the time of utility's peak demand, by strategic load shifting, is a viable option for industries to reduce their electricity cost. This paper presents an optimization model and formulation for load management for electrolytic process industries. The formulation utilizes mixed integer nonlinear programming (MINLP) technique for minimizing the electricity cost and reducing the peak demand, by rescheduling the loads, satisfying the industry constraints. The case study of a typical caustic-chlorine plant shows that a reduction of about 19% in the peak demand with a corresponding saving of about 3.9% in the electricity cost is possible with the optimal load scheduling under TOU tariff.

Designing globally optimal delta–sigma modulator topologies via signomial programming

AbstractWe present a design methodology for globally optimizing the topologies of delta–sigma modulators (DSMs). Previous work cast the design task into a general non‐convex, nonlinear programming problem, whereas we propose to recast it as a signomial programming problem. Convexification strategies are presented for transforming the signomial programming problem into its equivalent convex counterpart, thereby enabling the solution of globally optimal design parameters. It is also possible to include circuit non‐ideal effects that affect the transfer function of the modulator into the formulation without affecting the computational efficiency. The proposed framework has been applied to topology synthesis problems of single‐loop and multi‐loop low‐pass DSMs based on discrete‐time circuitry. Numerical results confirm the effectiveness of the proposed approach over conventional nonlinear programming techniques. Copyright © 2008 John Wiley & Sons, Ltd.

Optimization of Aircraft Wings for Gust Loads: Interval Analysis-Based Approach

An interval-based approach is presented for the optimization of aircraft structures under dynamic loads. As a specific application, the design is considered by including the effect of uncertainty present in the atmospheric turbulence and other parameters on the dynamic response of the aircraft wing structure. The stresses induced during a gust encounter are considered as the primary behavior constraints. The optimization procedure is illustrated with two examples. One is a symmetric double-wedge airfoil and the other is a supersonic airplane wing. The design parameters of the aircraft wing are assumed to be uncertain which are described as interval quantities. Interval analysis is used in the computation of the objective and constraint functions of the problem. An interval-based nonlinear programming technique is developed for the optimum solution of the two aircraft wings considered. The present methodology is expected to be more realistic compared with the probabilistic and fuzzy approaches in situations where either the probability distribution functions or the preference information for the uncertain design parameters is not available.

Optimization models for operative planning in drinking water networks

The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms, which in turn calls for carefully designed models with suitable properties. The paper develops a nonlinear mixed integer model and a nonlinear programming model with favorable properties for gradient-based optimization methods, based on smooth component models for the network elements. In combination with further nonlinear programming techniques (Burgschweiger et al. in ZIB Report ZR-05-31, Zuse Institute Berlin, 2005), practically satisfactory near-optimum solutions even for large networks can be generated in acceptable time using standard optimization software on a PC workstation. Such an optimization system is in operation at Berliner Wasserbetriebe.

Energy-Insensitive Guidance of Solid Motor Propelled Long Range Flight Vehicles Using MPSP and Dynamic Inversion

A energy-insensitive explicit guidance design is proposed in this paper by appending newly-developed nonlinear model predictive static programming technique with dynamic inversion, which render a closed form solution of the necessary guidance command update. The closed form nature of the proposed optimal guidance scheme suppressed the computational difficulties, and facilitate real-time solution. The guidance law is successfully verified in a solid motor propelled long range flight vehicle, for which developing an effective guidance law is more difficult as compared to a liquid engine propelled vehicle, mainly because of the absence of thrust cutoff facility. The scheme guides the vehicle appropriately so that it completes the mission within a tight error bound assuming that the starting point of the second stage to be a deterministic point beyond the atmosphere. The simulation results demonstrate its ability to intercept the target, even with an uncertainty of greater than 10% in the burnout time.

Multiobjective supplier selection using optimisation technique for balancing supply chain

This paper presents an innovative method to distribute sales volume among multiple suppliers in a multiobjective, multisource supplier selection scenario using optimisation techniques. The dynamics of assigning sales volume is captured in a mathematical model purported at satisfying multiple objectives. In such a multiobjective scenario, the theory of pareto optimality is introduced to formulate the process of optimal supplier selection. Linear Integer Programming method is proposed to solve the mathematical model for an optimal solution. The mathematical model is also retrofitted to solve for models with incremental discount pricing with the help of Non-Linear Programming techniques. The results illustrate the effectiveness and applicability of the technique under the discussed scenario. The results obtained using the linear integer programming method are also compared against a conventional process like Analytic Hierarchy Process and both the methods are appraised for their merits and demerits.

Two storage inventory model in a mixed environment

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters--holding costs, ordering costs, purchase costs, etc.--are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it's Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.

Second-order nonsmooth optimization for H ∞ synthesis

The standard way to compute H ∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H ∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.

Real-Coded Genetic Algorithm for Solving Generalized Polynomial Programming Problems

Generalized polynomial programming (GPP) is a nonlinear programming (NLP) method based on a nonconvex objective function, which is subject to nonconvex inequality constraints. Hence, a GPP problem has multiple local optima in its constrained solution space. General NLP techniques use local optimization, and therefore do not easily solve GPP problems. Some deterministic global optimization approaches have been developed to overcome this drawback of NLP methods. Although these approaches yield a global solution to a GPP problem, they can be mathematically tedious. Therefore, this study presents a real-coded genetic algorithm (RGA), which is a stochastic global optimization method, to find a global solution to a GPP problem. The proposed RGA is used to solve a set of GPP problems. The best solution obtained by the RGA is compared with the known global solution to each test problem. Numerical results show that the proposed RGA converges to a global solution to a GPP problem.

Applications of numerical optimal control to nonlinear hybrid systems

This paper develops a technique for numerically solving hybrid optimal control problems. The theoretical foundation of the approach is a recently developed methodology by S.C. Bengea and R.A. DeCarlo [Optimal control of switching systems, Automatica. A Journal of IFAC 41 (1) (2005) 11–27] for solving switched optimal control problems through embedding. The methodology is extended to incorporate hybrid behavior stemming from autonomous (uncontrolled) switches that results in plant equations with piecewise smooth vector fields. We demonstrate that when the system has no memory, the embedding technique can be used to reduce the hybrid optimal control problem for such systems to the traditional one. In particular, we show that the solution methodology does not require mixed integer programming (MIP) methods, but rather can utilize traditional nonlinear programming techniques such as sequential quadratic programming (SQP). By dramatically reducing the computational complexity over existing approaches, the proposed techniques make optimal control highly appealing for hybrid systems. This appeal is concretely demonstrated in an exhaustive application to a unicycle model that contains both autonomous and controlled switches; optimal and model predictive control solutions are given for two types of models using both a minimum energy and minimum time performance index. Controller performance is evaluated in the presence of a step frictional disturbance and parameter uncertainties which demonstrates the robustness of the controllers.

Nonlinear self-scheduling of a single unit small hydro plant in the day-ahead electricity market

This paper presents a model to solve the self- scheduling problem of a single unit small hydro plant, based on nonlinear programming techniques. The objective is to maximize the profit obtained by selling energy in the electricity market. The proposed model considers hydro power generation as a function of the water discharge and the net head and takes into account the unit start-ups and shut-downs. The scheduling problem is solved with commercially available optimization software. Input data are introduced through a worksheet, where results are exported once the optimization process is finished. Finally, results from a practical case study are further discussed.

Solving a dynamic separation problem using MINLP techniques

In the present paper a dynamic separation problem is modeled and solved using Mixed Integer Nonlinear Programming (MINLP) techniques. The objective is to maximize the profit for continuous cyclic operation, and at the same time, to find the optimal configuration for the separation column system. The dynamics of the chromatographic separation process are modeled as a boundary value problem which is solved, within the optimization, using an iterative finite difference method. The separation of a fructose-glucose mixture is solved using the Extended Cutting Plane (ECP) method. It is shown that the production planning can be done efficiently for different purity requirements, such that all the output of a system can be utilized. Using a process design that is optimized it is thus possible to use existing complex systems, or to design new systems, more efficiently and also to reduce the energy costs or the costs in general. The presented problem is a challenging application involving time-consuming calculations that, though the high capacity of todays computers, underlines the role of robust and fast numerical tools within optimization.