Optimality Conditions For Problems Research Articles

Optimal control of grinding mill circuit using model predictive static programming: A new nonlinear MPC paradigm

The recently developed reference-command tracking version of model predictive static programming (MPSP) is successfully applied to a single-stage closed grinding mill circuit. MPSP is an innovative optimal control technique that combines the philosophies of model predictive control (MPC) and approximate dynamic programming. The performance of the proposed MPSP control technique, which can be viewed as a ‘new paradigm’ under the nonlinear MPC philosophy, is compared to the performance of a standard nonlinear MPC technique applied to the same plant for the same conditions. Results show that the MPSP control technique is more than capable of tracking the desired set-point in the presence of model-plant mismatch, disturbances and measurement noise. The performance of MPSP and nonlinear MPC compare very well, with definite advantages offered by MPSP. The computational speed of MPSP is increased through a sequence of innovations such as the conversion of the dynamic optimization problem to a low-dimensional static optimization problem, the recursive computation of sensitivity matrices and using a closed form expression to update the control. To alleviate the burden on the optimization procedure in standard MPC, the control horizon is normally restricted. However, in the MPSP technique the control horizon is extended to the prediction horizon with a minor increase in the computational time. Furthermore, the MPSP technique generally takes only a couple of iterations to converge, even when input constraints are applied. Therefore, MPSP can be regarded as a potential candidate for online applications of the nonlinear MPC philosophy to real-world industrial process plants.

An Active Power Allocation Method for Wind-solar-batteries Hybrid Power System

In this article, an active power allocation method based on multi-objective optimization technique for wind-solar-batteries hybrid system (WSBHPS) is proposed. The problem of active power allocation of WSBHPS is formulated as following multi-objectives: minimizing of error between the output power of WSBHPS and desired power from the grid, minimizing start/stop switch number of power generation units, and maximizing the utilization of the regulation ability of wind turbines. The short-term power prediction of wind and solar power is utilized to quantify the available output power of each generation component, which is taken as constraint conditions of optimization problem with state of charge (SOC) of the batteries. The pattern-search algorithm is utilized to solve the multi-objective optimization problems. The proposed method is illustrated by implementing it to the active power allocation of a WSBHPS in simulation. The simulation results demonstrate that the proposed method has better active power output performance and less fluctuating rate compared with conventional average method. This method not only could maintain the SOC of batteries in an appropriate range, but also could reduce the on-or-off switching number of wind turbines and PV modules.

The turnpike property in finite-dimensional nonlinear optimal control

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory.

Augmented Lagrangian Duality for Composite Optimization Problems

In this paper, augmented Lagrangian duality is considered for composite optimization problems, and first- and second-order conditions for the existence of augmented Lagrange multipliers are presented. The analysis is based on the reformulation of the augmented Lagrangian in terms of the Moreau envelope functions and the technique of epi-convergence via the calculation of second-order epi-derivatives of the augmented Lagrangian. It is also proved that the second-order conditions for optimization problems with abstract constraints given in a form of set inclusions, obtained by Shapiro and Sun, can be derived directly from our general results.

On the equivalent static loads approach for dynamic response structural optimization

The equivalent static loads algorithm is an increasingly popular approach to solve dynamic response structural optimization problems. The algorithm is based on solving a sequence of related static response structural optimization problems with the same objective and constraint functions as the original problem. The optimization theoretical foundation of the algorithm is mainly developed in Park and Kang (J Optim Theory Appl 118(1):191---200, 2003). In that article it is shown, for a certain class of problems, that if the equivalent static loads algorithm terminates then the KKT conditions of the original problem and the final sub-problem are identical. The proof of this important theoretical result is unfortunately both incomplete and incorrect and the result is generally not valid. The missing parts of the proof are herein identified and corrected and the critical mistake in the proof is located and explained. We suggest a modified method in the same spirit for which the requested result is proved.

A novel fitness allocation algorithm for maintaining a constant selective pressure during GA procedure

During the evolution procedure of GA, the fitness distribution of population is always unforeseeable since it varies with many factors such as the nonlinearity and multimodality of optimization problem, crossover and mutation algorithms, and the progress of evolution procedure. For the GAs using stochastic selection mechanisms, fitness distribution sometimes significantly impacts on the performance of selection process, so as to make it very ineffective in protecting superior individuals and preventing inferior individuals. In this study, a new fitness scaling method, named powered distance sums scaling (PDSS), is proposed to eliminate the influence of fitness distribution on stochastic selection. Unlike previous approaches, the new method uses the powered sums of fitness distance to substitute for both raw fitness values and ranking numbers for computing scaled fitness. It maintains a much more constant and consistent selective pressure in different conditions of optimization problem and GA algorithm design, and may help GA designers in balancing exploration and exploitation during evolution procedures. Empirical studies are employed to illustrate the new method that through using the new scaling technique the convergence speed of GA search becomes more controllable.

Optimal portfolio choice for an insurer with loss aversion

The problem of optimal investment for an insurance company attracts more attention in recent years. In general, the investment decision maker of the insurance company is assumed to be rational and risk averse. This is inconsistent with non fully rational decision-making way in the real world. In this paper we investigate an optimal portfolio selection problem for the insurer. The investment decision maker is assumed to be loss averse. The surplus process of the insurer is modeled by a Lévy process. The insurer aims to maximize the expected utility when terminal wealth exceeds his aspiration level. With the help of martingale method, we translate the dynamic maximization problem into an equivalent static optimization problem. By solving the static optimization problem, we derive explicit expressions of the optimal portfolio and the optimal wealth process.

Analysis of fitness landscape modifications in evolutionary dynamic optimization

In this work, discrete dynamic optimization problems (DOPs) are theoretically analysed according to the modifications produced in the fitness landscape during the optimization process. Using the proposed analysis framework, the following DOPs are analysed: problems generated by the XOR DOP generator, three versions of the dynamic 0–1 knapsack problem, one problem involving evolutionary robots in dynamic environments, and the random dynamics NK-model. The XOR DOP generator creates benchmark DOPs from any binary static optimization problem, which allows to explore the properties of the static problem in a dynamic environment. Three types of transformations occurring in the fitness landscapes are observed in the DOPs analysed here. They are caused by: (i) permutation of solutions in the search space; (ii) duplication of solutions; and (iii) adding deviations to the fitness of a subset of solutions. The XOR DOP generator creates a special type of permutation that is not found in the other investigated DOPs. In this way, a new benchmark problem generator is proposed here based on the analysis performed, allowing to produce DOPs with six types of fitness landscape transformations, including those similar to the problems investigated in this paper. When compared to the XOR DOP generator, new algorithms can be tested and compared in a wider range of dynamic environments using the new generator. It is important to observe that some of the fitness transformations analysed here, like those caused by the duplication of solutions, are not currently explored in the evolutionary dynamic optimization area.

Equivalence between Neighboring-Extremal Control and Self-Optimizing Control for the Steady-State Optimization of Dynamical Systems

The problem of steering a dynamical system toward optimal steady-state performance is considered. For this purpose, a static optimization problem can be formulated and solved. However, because of uncertainty, the optimal steady-state inputs can rarely be applied directly in an open-loop manner. Instead, plant measurements are typically used to help reach the plant optimum. This paper investigates the use of optimizing control techniques for input adaptation. Two apparently different techniques of enforcing steady-state optimality are discussed, namely, neighboring-extremal control and self-optimizing control based on the null-space method. These two techniques are compared for the case of unconstrained real-time optimization in the presence of parametric variations. It is shown that, in the noise-free scenario, the two methods can be made equivalent through appropriate tuning. Note that both approaches can use measurements that are taken either at successive steady-state operating points or during the transient behavior of the plant. Implementation of optimizing control is illustrated through a simulated continuously stirred tank reactor (CSTR) example.

A hyper-heuristic based framework for dynamic optimization problems

Most of the real world problems have dynamic characteristics, where one or more elements of the underlying model for a given problem including the objective, constraints or even environmental parameters may change over time. Hyper-heuristics are problem-independent meta-heuristic techniques that are automating the process of selecting and generating multiple low-level heuristics to solve static combinatorial optimization problems. In this paper, we present a novel hybrid strategy for applicability of hyper-heuristic techniques on dynamic environments by integrating them with the memory/search algorithm. The memory/search algorithm is an important evolutionary technique that have applied on various dynamic optimization problems. We validate performance of our method by considering both the dynamic generalized assignment problem and the moving peaks benchmark. The former problem is extended from the generalized assignment problem by changing resource consumptions, capacity constraints and costs of jobs over time; and the latter one is a well-known synthetic problem that generates and updates a multidimensional landscape consisting of several peaks. Experimental evaluation performed on various instances of the given two problems validates that our hyper-heuristic integrated framework significantly outperforms the memory/search algorithm.

Векторна параметрична оптимізація робастних систем стабілізації інформаційно-вимірювальних пристроїв наземних рухомих об’єктів

Розглянуто особливості постановки задачі векторної параметричної оптимізації робастних систем стабілізації з урахуванням двох груп обмежень: умов стійкості системи та виконання надаваних до неї технічних вимог. Здійснено векторизацію критерію оптимізації робастної системи стабілізації інформаційно-вимірювальних пристроїв, експлуатованих на рухомих об’єктах. Запропоновано інтерактивну евристичну процедуру розв’язання цієї задачі, що складається з двох етапів. Ефективність запропонованої процедури підтверджено результатами моделювання

Neural Dynamics and Newton–Raphson Iteration for Nonlinear Optimization

In this paper, a special type of neural dynamics (ND) is generalized and investigated for time-varying and static scalar-valued nonlinear optimization. In addition, for comparative purpose, the gradient-based neural dynamics (or termed gradient dynamics (GD)) is studied for nonlinear optimization. Moreover, for possible digital hardware realization, discrete-time ND (DTND) models are developed. With the linear activation function used and with the step size being 1, the DTND model reduces to Newton–Raphson iteration (NRI) for solving the static nonlinear optimization problems. That is, the well-known NRI method can be viewed as a special case of the DTND model. Besides, the geometric representation of the ND models is given for time-varying nonlinear optimization. Numerical results demonstrate the efficacy and advantages of the proposed ND models for time-varying and static nonlinear optimization.

Cluster-based differential evolution with Crowding Archive for niching in dynamic environments

Real world optimization problems may very often be dynamic in nature, i.e. the position or height of the optima may change over time instead of being fixed as for static optimization problems. Dynamic Optimization Problems (DOPs) can pose serious challenges to the evolutionary computing community, especially when the search space is multimodal with multiple, time-varying optima. Some recent experimental studies have indicated that the process of evolutionary optimization can benefit from locating and tracking of several local and global optima instead of the single global optimum. This necessitates the integration of specially tailored niching techniques with an Evolutionary Algorithm (EA) for grouping of similar individuals in optimal basins of the landscape against drift and other disruptive forces as well as for making such individuals track the basins whenever dynamic changes appear. Motivated by such requirements, we present a multipopulation search technique involving a clustering strategy coupled with the memory-based Crowding Archive for dynamic niching in non-stationary environments. The algorithm uses Differential Evolution (DE) as its basic optimizer and is referred here as the Cluster-based DE with Crowding Archive (CbDE-wCA). It is equipped with a few robust strategies like favorable solution retention and generation, clearing strategy to eliminate redundant solutions, and crowding to restrict individuals to local search. The performance of the proposed algorithm has been tested on two different instances of the Moving Peaks Benchmark (MPB) problems. Experimental results indicate that CbDE-wCA can outperform other state-of-art dynamic multimodal optimizers in a statistically significant way, thereby proving its worth as an attractive alternative for niching in dynamic environments.

Trajectory Evaluation of Rotor-Flying Robots Using Accurate Inverse Computation Based on Algorithm Differentiation

Autonomous maneuvering flight control of rotor-flying robots (RFR) is a challenging problem due to the highly complicated structure of its model and significant uncertainties regarding many aspects of the field. As a consequence, it is difficult in many cases to decide whether or not a flight maneuver trajectory is feasible. It is necessary to conduct an analysis of the flight maneuvering ability of an RFR prior to test flight. Our aim in this paper is to use a numerical method called algorithm differentiation (AD) to solve this problem. The basic idea is to compute the internal state (i.e., attitude angles and angular rates) and input profiles based on predetermined maneuvering trajectory information denoted by the outputs (i.e., positions and yaw angle) and their higher-order derivatives. For this purpose, we first present a model of the RFR system and show that it is flat. We then cast the procedure for obtaining the required state/input based on the desired outputs as a static optimization problem, which is solved using AD and a derivative based optimization algorithm. Finally, we test our proposed method using a flight maneuver trajectory to verify its performance.

An improved firefly algorithm for solving dynamic multidimensional knapsack problems

There is a wide range of publications reported in the literature, considering optimization problems where the entire problem related data remains stationary throughout optimization. However, most of the real-life problems have indeed a dynamic nature arising from the uncertainty of future events. Optimization in dynamic environments is a relatively new and hot research area and has attracted notable attention of the researchers in the past decade. Firefly Algorithm (FA), Genetic Algorithm (GA) and Differential Evolution (DE) have been widely used for static optimization problems, but the applications of those algorithms in dynamic environments are relatively lacking. In the present study, an effective FA introducing diversity with partial random restarts and with an adaptive move procedure is developed and proposed for solving dynamic multidimensional knapsack problems. To the best of our knowledge this paper constitutes the first study on the performance of FA on a dynamic combinatorial problem. In order to evaluate the performance of the proposed algorithm the same problem is also modeled and solved by GA, DE and original FA. Based on the computational results and convergence capabilities we concluded that improved FA is a very powerful algorithm for solving the multidimensional knapsack problems for both static and dynamic environments.

Complex strength-constrained topology structural optimization problem statement for rolling stock and special equipment of railway

Purpose. The main paper purpose is the development of the topology structural optimization scientific basis regarding to the complicated optimization problems of rolling stock and special railway equipment structures. Methodology. Mathematical programming and mathematical modeling are the creating tools for the topology structural optimization problem statement for the rolling stock and special railway equipment. Findings. The fundamental review and analysis of the topology structural optimization modern state is executed. The classical variation problem statement and FE-statement of the topology optimization problem are in the paper. The stress-constrained structure mass minimization problem statement is considered. The stress-constrained topology optimization problems have some difficulties, which are considered in the paper in detail. The strength condition by the fatigue strength safety factor criterion is transformed to the strength condition by the allowable stresses criterion. Originality. Scientific novelty is the development of the optimal design theory adapted to solving the rolling stock and special railway equipment structures problems. Practical value. Practical importance of the research is the adaptation of the existing topology structural optimization problem statements to the railway engineering industry problems.

Use of Transient Measurements for the Optimization of Steady-State Performance via Modifier Adaptation

Real-time optimization (RTO) methods use measurements to offset the effect of uncertainty and drive the plant to optimality. RTO schemes differ in the way measurements are incorporated in the optimization framework. Explicit RTO schemes solve a static optimization problem repeatedly, with each iteration requiring transient operation of the plant to steady state. In contrast, implicit RTO methods use transient measurements to bring the plant to steady-state optimality in a single iteration, provided the set of active constraints is known. This paper considers the explicit RTO scheme “modifier adaptation” (MA) and proposes a framework that allows using transient measurements for the purpose of steady-state optimization. It is shown that convergence to the plant optimum can be achieved in a single transient operation provided the plant gradients can be estimated accurately. The approach is illustrated through the simulated example of a continuous stirred-tank reactor. The time needed for convergence is of th...

Modification of ABC Algorithm for Adaptation to Time‐Varying Functions

SUMMARYAlthough the ABC (artificial bee colony) algorithm is an effective tool for finding solutions to static optimization problems, application to dynamical problems that include a time‐varying element has not been assumed. A modification of the ABC algorithm for adaptation to time‐varying functions is proposed. To adjust to the change in the function, a procedure for reevaluating the bees at each time is introduced. It is shown that the proposed modification does not influence the search performance of the conventional algorithm. The efficiency of the modified ABC algorithm is demonstrated and verified through numerical simulations. © 2013 Wiley Periodicals, Inc. Electron Comm Jpn, 96(11): 44–53, 2013; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ecj.11479

Available Bandwidth Estimation for Network Selection in Heterogeneous Environment of WLAN and WiMAX

Always best connected services (ABC) allow multimode mobile terminals to stay connected to the best available networks, at anytime according to user preferences. One of the key aspects in realizing such ABC service is mainly attributed to an effective and dynamic access network selection process. However, most of the previous works consider the access network selection process as a static optimization problem which fails to address the dynamic QoS conditions intrinsic in wireless networks. One of the main challenges remaining is to find an efficient way of obtaining dynamic QoS parameters such as available bandwidth. In this paper, propose a novel dynamic access network selection algorithm capable of adapting to prevailing network conditions. Proposed algorithm is an estimation process where network selection in heterogeneous wireless environment (WiMAX & WLAN) performed using available bandwidth estimation relies on dynamic parameters designed using .net.

Node Positioning Algorithm in a Wireless Sensor Network

Node positioning technology in the Wireless sensor network is a very important issue. This article introduces the ant colony algorithm and its characteristics. It is a good solution to the static and dynamic portfolio optimization problem. The node positioning method in the wireless sensor network can be divided into two categories based ranging and no ranging. In order to get a higher positioning accuracy, and a further improvement the positioning accuracy, this paper introduces the ant colony optimization algorithm based on the first positioning method of the feasibility. The experimental results demonstrate the decrease of the number of transmitting beacon and the optimal mobile path after being optimized.