Nonlinear Inequality Constraints Research Articles

Global Nonparametric Sufficient Optimality Conditions for Semi-Infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η, ρ)-Invex Functions

In this paper, we establish a set of necessary optimality conditions and discuss a fairly large number of sets of global nonparametric sufficient optimality criteria under various generalized (η, ρ)-invexity assumptions for a minmax fractional programming problem with infinitely many nonlinear inequality and equality constraints.

PWM optimisation for three-level voltage inverter based on clonal selection algorithm

A clonal selection algorithm (CSA) is first applied to determine the switching time for a three-level voltage pulse width modulation (PWM) inverter. To obtain an optimal PWM switching pattern, CSA should calculate a set of nonlinear transcendental equations with nonlinear equality and inequality constraints. Unlike the general CSA by penalising infeasible solutions, this new CSA deals with constraints and guides the search to the true optimum solution by an infeasibility degree with a random disturbance (IFDD) selection operation. The proposed method minimise the total harmonic distortion and generates a waveform with adjustable amplitude of the fundamental component. Calculation results verify that the proposed method is efficient to obtain the optimal PWM switching pattern and is superior to natural sampling and genetic algorithm-based PWM switching pattern.

Augmented Lagrangian Particle Swarm Optimization in Mechanism Design

The problem of optimizing nonlinear multibody systems is in general nonlinear and nonconvex. This is especially true for the dimensional synthesis process of rigid body mechanisms, where often only local solutions might be found with gradient-based optimization methods. An attractive alternative for solving such multimodal optimization problems is the Particle Swarm Optimization (PSO) algorithm. This stochastic solution technique allows a derivative-free search for a global solution without the need for any initial design. In this work, we present an extension to the basic PSO algorithm in order to solve the problem of dimensional synthesis with nonlinear equality and inequality constraints. It utilizes the Augmented Lagrange Multiplier Method in combination with an advanced non-stationary penalty function approach that does not rely on excessively large penalty factors for sufficiently accurate solutions. Although the PSO method is even able to solve nonsmooth and discrete problems, this augmented algorithm can additionally calculate accurate Lagrange multiplier estimates for differentiable formulations, which are helpful in the analysis process of the optimization results. We demonstrate this method and show its very promising applicability to the constrained dimensional synthesis process of rigid body mechanisms.

New Sequential Quadratically-Constrained Quadratic Programming Method of Feasible Directions and Its Convergence Rate

This paper discusses optimization problems with nonlinear inequality constraints and presents a new sequential quadratically-constrained quadratic programming (NSQCQP) method of feasible directions for solving such problems. At each iteration. the NSQCQP method solves only one subproblem which consists of a convex quadratic objective function, convex quadratic equality constraints, as well as a perturbation variable and yields a feasible direction of descent (improved direction). The following results on the NSQCQP are obtained: the subproblem solved at each iteration is feasible and solvable: the NSQCQP is globally convergent under the Mangasarian-Fromovitz constraint qualification (MFCQ); the improved direction can avoid the Maratos effect without the assumption of strict complementarity; the NSQCQP is superlinearly and quasiquadratically convergent under some weak assumptions without thestrict complementarity assumption and the linear independence constraint qualification (LICQ).

Impact of Fault Level Constraints on the Economic Operation of Power Systems

The aim of this paper is to prove that fault levels may have a significant impact on the economic operation of modern power systems. First, we present a simple method for the incorporation of fault level constraints (FLCs) in the optimal power flow (OPF), the main optimization tool for the economic operation of power systems. The constraints imposed by fault levels are converted to simple nonlinear (inequality) constraints, described by variables of the conventional OPF. Most common OPF-solving engines already have the computational capacity to handle numerous nonlinear constraints, such as the ones described by the power balance equations on buses. Therefore, once FLCs are converted to nonlinear constraints described by OPF variables, they can be directly introduced to any optimization process performing the OPF. We applied this enhanced OPF on a simple 12-bus network. The results verified the significant impact fault levels have on the optimum operating point of the power system

Online Minimum-Copper-Loss Control of an Interior Permanent-Magnet Synchronous Machine for Automotive Applications

This paper presents an algorithm to calculate the current references on line, considering the inherent nonlinear nature of the saturation effect in an interior permanent-magnet synchronous machine for the minimum-copper-loss control under the current and voltage limit of the drive system. This paper basically approaches this issue as a nonlinearly constrained optimization problem where the torque command imposes the nonlinear equality constraint and the voltage limit imposes the nonlinear inequality constraint. Depending on the operating region, it solves the corresponding set of nonlinear equations in real time derived from the Lagrange multiplier method. Newton's method among various techniques is adopted to implement the numerical solution. This scheme gives accurate results not only in motoring but also in generating operation of the machine, since the voltage drop of the stator resistance is taken into account, which is hardly applicable to a two-dimensional lookup table where the inputs are the torque command and the maximum flux amplitude, and the output is each axis current reference in the rotor reference frame. The simulation and experimental results show the feasibility and performance of the proposed technique

An active set sequential quadratic programming algorithm for nonlinear optimisation

In this paper, we have proposed an active set feasible sequential quadratic programming algorithm for nonlinear inequality constraints optimization problems. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving a reduced quadratic programming subproblem. To overcome the Maratos effect, a higher-order correction direction is obtained by solving a reduced least square problem. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without strict complementarity.

A strongly convergent norm-relaxed method of strongly sub-feasible direction for optimization with nonlinear equality and inequality constraints

In this paper, a class of optimization problems with nonlinear equality and inequality constraints is discussed. Firstly, the original problem is transformed to an associated simpler auxiliary optimization problem with only inequality constraints and a penalty parameter, and the later problem is showed to be equivalent to the original problem if the parameter is large enough (but finite). Then, combining the norm-relaxed Method of Feasible Direction (MFD) with the idea of Method of Strongly Sub-Feasible Direction (MSSFD), we present an algorithm with arbitrary initial point for the original problem. At each iteration of the auxiliary problem, an improved search direction is obtained by solving one Direction Finding Subproblem (DFS), i.e., a quadratic program, which always possesses a solution. In the process of iteration, the feasibility of the iteration points is monotone increasing. Furthermore, whenever an iteration point enters the feasible set, the proposed algorithm reduces to a feasible and decent method for the auxiliary problem. Under some suitable assumptions, the global and strong convergence of the proposed algorithm can be obtained. Finally, some elementary numerical experiments are reported.

Maximum Likelihood Inference for Log-linear Models Subject to Constraints of Double Monotone Dependence

To model an hypothesis of double monotone dependence between two ordinal categorical variables A and B usually a set of symmetric odds ratios defined on the joint probability function is subject to linear inequality constraints. Conversely in this paper two sets of asymmetric odds ratios defined, respectively, on the conditional distributions of A given B and on the conditional distributions of B given A are subject to linear inequality constraints. If the joint probabilities are parameterized by a saturated log-linear model, these constraints are nonlinear inequality constraints on the log-linear parameters. The problem here considered is a non-standard one both for the presence of nonlinear inequality constraints and for the fact that the number of these constraints is greater than the number of the parameters of the saturated log-linear model.

Likelihood Ratio Test for and Against Nonlinear Inequality Constraints

In applied statistics a finite dimensional parameter involved in the distribution function of the observed random variable is very often constrained by a number of nonlinear inequalities. This paper is devoted to studying the likelihood ratio test for and against the hypothesis that the parameter is restricted by some nonlinear inequalities. The asymptotic null distributions of the likelihood ratio statistics are derived by using the limits of the related optimization problems. The author also shows how to compute critical values for the tests.

Nonlinear Trajectory Generation for Autonomous Vehicles via Parametrized Maneuver Classes

A technique is presented for creating continuously parameterized classes of feasible system trajectories. These classes, which are useful for higher-level vehicle motion planners, follow directly from a small collection of userprovided example motions. A dynamically feasible trajectory interpolation algorithm generates a continuous family of vehicle maneuvers across a range of boundary conditions while enforcing nonlinear system equations of motion as well as nonlinear equality and inequality constraints. The scheme is particularly useful for describing motions that deviate widely from the range of linearized dynamics and where satisfactory example motions may be found from off-line nonlinear programming solutions or motion capture of human-piloted flight. The interpolation algorithm is computationally efficient, making it a viable method for real-time maneuver synthesis, particularly when used in concert with a vehicle motion planner. Experimental application to a three-degree-of-freedom rotorcraft test bed demonstrates the essential features of system and trajectory modeling, maneuver example selection, maneuver class synthesis, and integration into a hybrid system path planner.

Direct Incorporation of Fault Level Constraints in Optimal Power Flow as a Tool for Network Capacity Analysis

The aim of this paper is to present a method for the direct incorporation of fault level constraints (FLCs) in the optimal power flow (OPF) as a tool for network capacity analysis, i.e., optimal generation expansion planning within an existing network. A mathematical methodology to convert constraints imposed by fault levels to simple nonlinear inequality constraints is developed. No new variables are introduced in the OPF formulation to describe the additional constraints. Most common OPF-solving engines already have the computational capacity to handle numerous nonlinear constraints, such as the ones described by the power balance equations on buses. Therefore, once FLCs are converted to nonlinear constraints described by OPF variables, they can be directly introduced to any optimization process performing the OPF. A 12-bus/15-line test case demonstrates the advantages of the new method in comparison with a previously proposed iterative method that converted them to restrictions on new capacity. It also proves that when FLCs are ignored, the capacity of the network to absorb new generation is overestimated.

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.

Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization

The orthogonal packing of rectangular items in an arbitrary convex region is considered in this work. The packing problem is modeled as the problem of deciding for the feasibility or infeasibility of a set of nonlinear equality and inequality constraints. A procedure based on nonlinear programming is introduced and numerical experiments which show that the new procedure is reliable are exhibited. Scope and purpose We address the problem of packing orthogonal rectangles within an arbitrary convex region. We aim to show that smooth nonlinear programming models are a reliable alternative for packing problems and that well-known general-purpose methods based on continuous optimization can be used to solve the models. Numerical experiments illustrate the capabilities and limitations of the approach.

Self-Adaptive Fitness Formulation for Evolutionary Constrained Optimization of Water Systems

In design of water distribution networks, there are several constraints that need to be satisfied; supplying water at an adequate pressure being the main one. In this paper, a self-adaptive fitness formulation is presented for solving constrained optimization of water distribution networks. The method has been formulated to ensure that slightly infeasible solutions with a low objective function value remain fit. This is seen as a benefit in solving highly constrained problems that have solutions on one or more of the constraint bounds. In contrast, solutions well outside the constraint bounds are seen as containing little genetic information that is of use and are therefore penalized. In this method, the dimensionality of the problem is reduced by representing the constraint violations by a single infeasibility measure. The infeasibility measure is used to form a two-stage penalty that is applied to infeasible solutions. The performance of the method has been examined by its application to two water distribution networks from literature. The results have been compared with previously published results. It is shown that the method is able to find optimum solutions with less computational effort. The proposed method is easy to implement, requires no parameter tuning, and can be used as a fitness evaluator with any evolutionary algorithm. The approach is also robust in its handling of both linear and nonlinear equality and inequality constraint functions. Furthermore, the method does not require an initial feasible solution, this being an advantage in real-world applications having many optimization variables.

An SQP feasible descent algorithm for nonlinear inequality constrained optimization without strict complementarity

In this paper, a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed, and a new SQP feasible descent algorithm for solving the problems is presented. At each iteration of the new algorithm, a convex quadratic program (QP) which always has feasible solution is solved and a master direction is obtained, then, an improved (feasible descent) direction is yielded by updating the master direction with an explicit formula, and in order to avoid the Maratos effect, a height-order correction direction is computed by another explicit formula of the master direction and the improved direction. The new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Furthermore, the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted. Finally, some numerical tests are reported.

モデル誤差を伴うシステムにおける最適コストの評価手法

At the phase of feasibility study on the system for process optimization, effectiveness of the system is often estimated in terms of a mathematical program formulated from the regional analysis of the operating results. A two-level mathematical programming is presented, which predicts the optimal cost in the worst case caused by the error involved in the process model. In particular, nonlinear inequality constraints of the upper level problem are proposed to give a limit on the acceptable error range. In order to examine the efficiency of the proposed method, a case study on the operation optimizing system of the electric power plant is reported.

Constrained nonlinear optimization for solubility parameters of poly(lactic acid) and poly(glycolic acid)—validation and comparison

An optimization technique has been proposed to determine Hansen solubility parameters (HSP) and radius of interaction (R) of PLA and PGA; objective function to be minimized being radius of interaction (R), and the constraints being that the solvents should be within and nonsolvents outside the interaction sphere. The proposed method has been validated and found to be most reliable. The values of HSP (δd, δp, δh) and R for PLA have been obtained as ((18.50, 9.70, 6.00) and 10.50)(J/cc)0.5 at 25°C; and those of PGA as ((17.70, 6.21, 12.50) and 1.92)(J/cc)0.5 at 80°C. For formulating the nonlinear inequality constraints known HSP data for 20 solvents and seven nonsolvents have been used for PLA at 25°C; similarly HSP data for three newly found solvents (phenol, m-cresol and 4-chlorophenol), and five nonsolvents have been used for PGA at 80°C. Established methods have been used for comparison. HSP and R have been directly compared using the 3D intrinsic viscosity and classical methods. Indirectly the total solubility parameter δ has been compared with the values obtained from the intrinsic viscosity 1D approach and group contribution method using Fedors and van Krevelen correlations. The 1D approach has led to an empirical correlation for intrinsic viscosities of PLA and PGA.

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR MINMAX FRACTIONAL OPTIMAL CONTROL PROBLEMS CONTAINING ARBITRARY NORMS

Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

A Recurrent Neural Network for Nonlinear Convex Optimization Subject to Nonlinear Inequality Constraints

This paper presents a novel recurrent neural network for solving nonlinear convex programming problems subject to nonlinear inequality constraints. Under the condition that the objective function is convex and all constraint functions are strictly convex or that the objective function is strictly convex and the constraint function is convex, the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution. Compared with the existing neural networks for solving such nonlinear optimization problems, the proposed neural network has two major advantages. One is that it can solve convex programming problems with general convex inequality constraints. Another is that it does not require a Lipschitz condition on the objective function and constraint function. Simulation results are given to illustrate further the global convergence and performance of the proposed neural network for constrained nonlinear optimization.