Geometric based estimation and nonlinear PI controller for dynamic optimization problem

Solving online dynamic time-linkage problems under unreliable prediction

Dynamic multiobjective optimization problem (DMOP) is a class of complex dynamic optimization problem (DOP), its widely exists in many real-world problems. Firstly, an core estimation of distribution model which used to approximately estimate the Pareto solution in the next environment is given, when a change in the environment is detected, the algorithm uses the collected information from the previous searching environments to predict the location of the Pareto solution in the next environment. Furthermore, a dynamic multiobjective optimization evolutionary algorithm with core estimation of distribution is proposed. The computer simulations are made on two dynamic multiobjective optimization problems, and the results indicate the proposed algorithm is effective.

Optimization issues are often defined statically, assuming constant environmental conditions.However, in many real-world scenarios, problem environments are dynamic and continuously changing.Thus we need optimization algorithms that could solves those issues in dynamic environments as well.Dynamic optimization problems are change(s) that may occur through the time.Such environments are characterized by uncertainty, temporal changes, and structural complexities, which makes the optimization process a significant challenge.In addressing these challenges, evolutionary algorithms have emerged as one of the most effective approaches for solving dynamic optimization problems (DOPs).Among these algorithms, the Imperialist Competitive Algorithm (ICA), designed based on swarm intelligence and competition among imperialist countries, has garnered considerable attention due to its capability in solving static optimization issues.In this research, Imperialist Competitive Algorithms, inspired by the historical and political processes of colonization and assimilation, have been known as one of the efficient evolutionary algorithms.These algorithms face numerous challenges when dealing with dynamic problems , including reduced population diversity, performance degradation in conditions of rapid environmental changes, and limitations in optimal convergence.These cases indicate the need to develop improved and more adaptable versions of these algorithms.Using concepts such as memory, population clustering, and repulsion mechanisms, this algorithm has been able to maintain population diversity at all stages while increasing the speed of convergence in the face of environmental changes.The key feature of the proposed algorithm is the use of memory to store previous optimal solutions, a clustering mechanism to manage population diversity, and repulsion to prevent unnecessary accumulation of solutions in specific regions.Nevertheless, ICA exhibits poor performance in dynamic environments because it lacks mechanisms to maintain diversity, quick adaptation to environmental changes, and new optima track.This study presents an improved version of ICA aimed at overcoming these limitations.The proposed algorithm incorporates a combination of a memory mechanism and a clustering strategy to enhance its adaptability to environmental changes and preserve diversity within the population.The memory mechanism stores information about previous optima and utilizes it under appropriate conditions to accelerate the optimization process.forclustering method is used for clustering.Clustering in the proposed method ensures that diversity is maintained for the population during the execution of the algorithm.In this study, our goal is to solve problems that change the environment in a global way.That means, the fitness of all points in the environment changes.By testing just one point in the environment and comparing the fitness obtained with its previously stored value, we can detect a change in the environment.On the other hand, the clustering strategy, particularly the k-means technique, to maintain maintain population diversity and prevents the convergence of solutions to specific regions.Together, these two components create a balance between exploration and exploitation, thereby improving the algorithm's performance in dynamic environments.To evaluate the performance of the proposed algorithm, the Moving Peaks Benchmark (MPB) was used as a standard metric.Due to its capability to simulate complex and diverse changes in dynamic environmentsparticularly in Branke's second scenario-MPB is one of the most recognized tools for assessing the performance of dynamic optimization algorithms.The proposed algorithm was evaluated alongside advanced algorithms such as FTmPSO (TMO), RAmQSO-s4, RmNAFSA-s4, TFTmPSO, RFTmPSO, mQSO10 (5+5q), FMSO, CellularPSO, Multi-SwarmPSO, mCPSO, AmQSO*, FTMPSO, almPSO, and CDEPSA.Experimental results demonstrated that the proposed algorithm outperformed other methods in areas such as convergence speed, adaptability to environmental changes, and population diversity preservation.A key feature of the proposed algorithm is its ability to retain identified optima even after environmental changes.Additionally, the use of the k-means clustering technique has ensured that the algorithm effectively avoids excessive focus on specific regions and maintains population diversity while facing complex environmental changes.Another advantage of this algorithm is its scalability in handling dynamic optimization issues with high dimensions and complexities.These findings indicate that the proposed algorithm is not only effective in laboratory settings but also suitable for real-world applications with fast and dynamic changes.

Quantum particles-enhanced multiple Harris Hawks swarms for dynamic optimization problems

The electric energy consumption in passenger vehicles is rapidly increasing. To limit the associated increase in fuel consumption, an energy management system has been developed. This system exploits the fact that the losses in the internal combustion engine vary with the operating point, and uses the possibility to temporarily store electric energy in a battery, such that electric energy is only produced at moments when it is cheap to generate. To come to a practically applicable solution, a vehicle model is derived, containing only the component characteristics relevant for this application. The energy management problem is formulated as an optimization problem. The fuel consumption over a driving cycle is minimized, while respecting physical limitations of the components and maintaining an acceptable energy level of the battery. Several optimization methods are studied to come to a solution. The dynamic optimization problem is solved using Dynamic Programming. After rewriting it as a static optimization problem and approximating the cost function by a quadratic function, the problem is solved using Quadratic Programming, which requires less computation time. A real-time implementable strategy has been derived from the Quadratic Programming problem, that does not require a prediction of the future driving cycle. This strategy compares the current cost of producing electric energy with the estimated average cost. By adapting the average cost based on the energy level of the battery, it is ensured that the battery energy level will remain around the desired value. Simulations show that a fuel reduction up till 2% can be obtained on a conventional vehicle without major hardware changes. Higher reductions are possible on the exhaust emissions. To predict and explain the amount of fuel reduction that can be obtained with a given vehicle configuration, a set of engineering rules is derived based on typical component characteristics. Their results correspond reasonably well with the simulations. An advanced power net topology is studied which contains both a battery and an ultracapacitor that are connected by a DC-DC converter and a switch. Because of the increased complexity, this system is modeled using linear and piecewise linear approximations of the component characteristics, such that the energy management problem can be casted as a Linear Programming problem. The discrete switch makes it a Mixed Integer Linear Programming Problem. A realtime strategy, similar to the strategy for the conventional power net, has been derived. The addition fuel reduction that is obtained with the dual storage power net is small, because the maximum profit that can be obtained with an ideal lossless battery is not much higher than with a normal battery. Subsequently, hybrid electric vehicles are studied that use an Integrated Starter Generator which can be used for generating electric power and for vehicle propulsion. Several conSummary figurations are studied with respect to their potential fuel reduction. Configurations that enable start-stop operation of the engine obtain a much higher fuel reduction, up to 40%. The controllers are tested in real-time on a Hardware-in-the-Loop environment, where a vehicle simulation model is combined with existing electric components. It is shown that the electric power setpoints provided by an energy management strategy can be realized in practice.

Large scale optimization strategies have evolved considerably over the past two decades. Currently, continuous variable optimization problems (nonlinear programs) are solved on-line for steady state refinery models with several hundred thousand variables. Moreover, efficient NLP strategies have been developed for dynamic optimization problems. Still, to take the next step, on-line optimization of large dynamic chemical processes, a number of limitations and research challenges must be overcome. Many of the advances in NLP algorithms have taken place by recognizing and exploiting the framework of Successive Quadratic Programming (SQP) algorithms. These are extensions of Newton type methods for converging to the solution of the KKT (optimality) conditions of the optimization problem. Because of this, fast convergence can be expected and a number of standard devices can be added to stabilize the algorithm to converge from poor starting points. Limitations of these Newton-based methods are also well-known. They experience difficulties in the presence of ill conditioning and extreme nonlinearities. Also, for optimization algorithms, nonconvexity can also lead to a number of difficulties and there is a need for software that allows exploitable structures for specific problem classes. A number of innovations in algorithm design and problem formulation address these issues and greatly improve performance. As a result, very fast NLP algorithms can be derived for data reconciliation, parameter estimation, nonlinear model predictive control and dynamic optimization. Moreover, inequality constraints and variable bounds can be treated through advances in interior point strategies. These methods preserve the particular problem structure and scale well in performance for large-scale problems with many constraints. Finally, the ability to solve nonlinear programs quickly also allows us to consider more challenging problems. These include the extension to solving nonconvex NLPs globally and the ability to assess the solution’s tolerance to uncertainty by considering features of nonlinear programming sensitivity analysis and robust optimization through flexibility analysis and design.

The solution of a combined static and dynamic optimization problem is considered in this paper. By means of a feasible decomposition procedure this combined problem is first decomposed into a two-level optimization problem. In this new problem the first-level subproblem is found to be one of conventional optimal control and mathematical programming which can be solved using known techniques. For the solution of the second-level problem a computational algorithm is developed using the gradient projection method. The existence of the solution is investigated. To illustrate the applicability of the method, a simplified gantry crane yard design problem is considered and some computational results are given.

ABSTRACTThis article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.

Chapter 25 - Uncertain optimization (with a special focus on data envelopment analysis)

In this paper, online convex optimization (OCO) problems with time-varying objective and constraint functions are studied from the perspective of an agent who takes actions in real time. Information about the current objective and constraint functions is revealed only after the corresponding action is already chosen. Inspired by a fast converging algorithm for time-invariant optimization in the very recent work [1], we develop a novel online algorithm based on virtual queues for constrained OCO. Optimal points of the dynamic optimization problems with full knowledge of the current objective and constraint functions are used as a dynamic benchmark sequence. Upper bounds on the regrets with respect to the dynamic benchmark and the constraint violations are derived for the presented algorithm in terms of the temporal variations of the underlying dynamic optimization problems. It is observed that the proposed algorithm possesses sublinear regret and sublinear constraint violations, as long as the temporal variations of the optimization problems are sublinear, i.e., the objective and constraint functions do not vary too drastically across time. The performance bounds of the proposed algorithm are superior to those of the state-of-the-art OCO method in most scenarios. Besides, different from the saddle point methods widely used in constrained OCO, the stepsize of the proposed algorithm does not rely on the total time horizon, which may be unknown in practice. Finally, the algorithm is applied to a dynamic resource allocation problem in data center networks. Numerical experiments are conducted to corroborate the merit of the developed algorithm and its advantage over the state-of-the-art.

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min-max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.

When facing dynamic optimization problems the goal is no longer to find the extrema, but to track their progression through the space as closely as possible. Over these kind of over changing, complex and ubiquitous real-world problems, the explorative-exploitive subtle counterbalance character of our current state-of-the-art search algorithms should be biased towards an increased explorative behavior. While counterproductive in classic problems, the main and obvious reason of using it in severe dynamic problems is simple: while we engage ourselves in exploiting the extrema, the extrema moves elsewhere. In order to tackle this subtle compromise, we propose a novel algorithm for optimization in dynamic binary landscapes, stressing the role of negative feedback mechanisms. The Binary Ant Algorithm (BAA) mimics some aspects of social insects' behavior. Like Ant Colony Optimization (ACO), BAA acts by building pheromone maps over a graph of possible trails representing pseudo-solutions of increasing quality to a specific optimization problem. Main differences rely on the way this search space is represented and provided to the colony in order to explore/exploit it, while and more important, we enrol in providing strong evaporation to the problem-habitat. By a process of pheromone reinforcement and evaporation the artificial insect's trails over the graph converge to regions near the ideal solution of the optimization problem. Over each generation, positive feedbacks made available by pheromone reinforcement consolidate the best solutions found so far, while enhanced negative feedbacks given by the evaporation mechanism provided the system with population diversity and fast self-adaptive characteristics, allowing BAA to be particularly suitable for severe complex dynamic optimization problems. Experiments made with some well known test functions frequently used in the Evolutionary Algorithms' research field illustrate the efficiency of the proposed method. BAA was also compared with other algorithms, proving to be more able to track fast moving extrema on several test problems.

In this paper we propose a self adaptive cluster based Differential Evolution (DE) algorithm to solve the Dynamic Optimization Problems (DOPs). We have enhanced the classical DE to perform better in dynamic environments by a powerful clustering technique. During evolution, the information gained by the particles of different clusters is exchanged by a self adaptive strategy. The information exchange is done by re-clustering, and the cluster number is updated adaptively throughout the optimization process. To detect the environment change a test particle is used. Moreover, to adapt the population in new environment an External Archive is also used. The performance of SACDEEA is evaluated on GDBG benchmark problems and compared with other existing algorithms.KeywordsDifferential EvolutionDynamic Optimization Problem (DOP)Self adaptive clustering

The importance of optimal marketing communications mix decisions is well-recognized by both marketing scholars and practitioners. A significant volume of work has addressed the problem of dynamic marketing mix optimization assuming constant effectiveness of marketing instruments. However, the effectiveness of marketing communications varies over time for a variety of reasons. Moreover, due to factors such as inflation or deflation in media prices and/or raw material inputs, there can be differential changes in the costs of communications and/or margins on the good (or service) sold over time. The academic literature offers little normative direction on how time-varying marketing effectiveness and costs drive optimal marketing-mix levels and their relative allocation. The authors shed light on these issues by solving a monopoly firm's finite horizon dynamic marketing communications mix optimization problem involving two marketing instruments with time-varying parameters, i.e., the marketing effectiveness parameters, media costs, and product margin are all allowed to vary over time. First, they find that the structure of the solutions is similar to that of the classic Nerlove–Arrow model, for a completely general nature of time-varying effectiveness. Second, their model can be used by managers to exactly determine whether and when to switch their marketing-mix emphasis (defined by the marketing element receiving the dominant portion of the budget) over a finite planning horizon. In sum, the authors expand knowledge on optimal allocation of marketing resources with time-varying effectiveness. They also extend their solution to incorporate multiple (more than two) marketing instruments.

This paper proposes a solution method for the discrete-time long-term dynamic portfolio optimization problem with state and asset allocation constraints. We use the ideas of Model Predictive Control (MPC) to solve the constrained stochastic control problem. MPC is a solution technique which was developed to solve constrained optimal control problems for deterministic control applications. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic Gaussian processes. We prove that MPC is a suboptimal control strategy for stochastic systems which uses the new information advantageously and thus is better than the pure optimal open-loop control. For the open-loop optimal control optimization, we derive the conditional portfolio distribution and the corresponding conditional portfolio mean and variance. The mean and the variance depend on future decision about the asset allocation. For the dynamic portfolio optimization problem, we consider constraints on the asset allocation as well as probabilistic constraints on the attainable values of the portfolio wealth. We discuss two different objectives, a classical mean–variance objective and the objective to maximize the probability of exceeding a predetermined value of the portfolio. The dynamic portfolio optimization problem is stated, and the solution via MPC is explained in detail. The results are then illustrated in a case study.