Optimality Conditions For Problems Research Articles

Optimal Neumann control of the wave equation with L 1-control cost: the finite-time turnpike property

The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L 1 -norm control cost with a weight γ > 0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L 2 -norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L 1 -norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L 2 -sense.

Dynamic and static fusion mechanisms of infrared and visible images

This paper propose a dynamic fusion mechanism of infrared and visible images, named DIFM, capable of solving the static fusion optimization problem. The DIFM correlates the image fusion quality with the image restoration quality to construct a unified optimization loss function. According to the DIFM, a dynamic image fusion network of infrared and visible images is constructed and is therefore denoted with DF-Net. Specifically, the DF-Net comprises two modules, i.e., the dynamic fusion module (DFM) and the self-learning dynamic restoration module (SLDRM). In order to solve the static fusion problem of existing methods, the DFM is proposed to learn the fusion weight dynamically. Specifically, the DFM comprises a classification module (CM) and an image fusion module (IFM), which determine whether and how to fuse source images. In addition, a unified fusion loss function is introduced to obtain more hidden features of infrared and visible images in complex environments. Therefore, the stumbling block of deep learning in image fusion, i.e., static fusion, is significantly mitigated. Extensive experiments demonstrate that the dynamic fusion optimization method neatly outperforms the state-of-the-art methods in most metrics.

Optimal control problem with state constraints via penalty functions

In this paper, we derive necessary conditions of optimality for problems with state constraints using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., de Pinho et al., 2022). Our aim is to provide a proof of the Maximum Principle that can be easily followed by those with basic knowledge of classical optimal control and elementary functional analysis. We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact.

Multiple landscape measure-based approach for dynamic optimization problems

Many practical decision-making problems involve dynamic scenarios, where the decision variables, conditions and/or parameters of their optimization models change over time. Such problems are known as dynamic optimization problems (DOPs). Although evolutionary algorithms (EAs) have been effective in solving static optimization problems, they face challenges in handling DOPs. To improve the performance of EAs in dealing with DOPs, this paper proposes a new evolutionary framework that uses different landscape measures to analyze problem landscapes and utilizes the information gained from this to improve the searching process. The proposed method is adopted in three EAs to deal with dynamic functions from IEEE-CEC2009 and two real-world problems. According to the experimental results, LIDOA with multi-measure methods improves the performance of GA, jDE and CMA-ES, on average by 6.71%, 3.03% and 7.78% on benchmark problems, respectively.

A Cooperative Evolutionary Computation Algorithm for Dynamic Multiobjective Multi-AUV Path Planning

Multiple autonomous underwater vehicles (AUVs) are popular for executing submarine missions, which involve multiple targets distributed in a large and complex underwater environment. The path planning of multiple AUVs is a significant and challenging problem, which determines the location of surface points for AUV launch and plans the paths of AUVs for target traveling. Most existing works model the problem as a single-objective static optimization problem. However, the target missions may change over time, and multiple optimization objectives are usually expected for decision making. Thus, this paper models the problem as a dynamic multiobjective optimization problem and proposes a cooperative evolutionary computation algorithm to provide diverse and high-quality solutions for decision makers. In the proposed method, solutions are represented using a bi-layer encode scheme, in which the first layer indicates the surface location points and the second layer represents the traveling sequences of target missions. Multiple populations for multiple objectives (MPMO) framework is adopted to efficiently solve the dynamic multiobjective AUV optimization problem. In addition, a recombination-based sampling strategy is developed to improve convergence by fusing the information of multiple populations. Once a change occurs, an incremental response strategy is adopted to generate high-quality solutions for population evolution. Based on the dataset of New Zealand bathymetry, six complex underwater scenarios are constructed with a size of 50 km×50 km×10 km and 400 target missions for tests. Experimental results show that the proposed method outperforms the state-of-the-art algorithms in terms of solution diversity and optimality.

The link between MiFID and Risk Appetite Framework as an application of best practices for wealth management and the entire value chain of the financial industry

After a short review of the MiFID regulations and the RAF, the paper identifies the link between them which allows to mitigate a balance sheet risk sustained by the financial intermediary and, at the same time, to improve its stability and value creation, through a maximization of customer loyalty. The client’s attitude towards risk can be summarized in these terms: "I don't like risk, but I like to win". Thus, a three-dimensional approach towards expected utility is suggested for estimating risk tolerance: risk aversion, loss aversion and reflection. In addition, a definition of the client's financial objectives is required, combined with greater disclosure - which allows the construction of a financial statement – to attest that risk-taking is indeed a luxury, as indicated by the metrics of the discretionary wealth ratio, and therefore, of the Standard of Living Risk (SLR). The next step consists in the determination of a set of portfolios along the efficient frontier where risk is represented by the expected shortfall, the determination of which belongs to a Generalized Extreme Value Theory logic. The client's objectives are described in terms of probability of success, where the latter is a function of an initial endowment, a potential positive contribution of financial resources over time as well as an expected return level. The above is expressed through a practical case that envisages the determination of a set of EGPF portfolios and the identification of the specific portfolio, obtained as a solution to a static and dynamic optimization problem, where the objectives have been formalized through the calculation of the associated utility

An aspect of the turnpike property. Long time horizon behavior

The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval.
 In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

Orlicz risks for assessing stochastic streamflow environments: a static optimization approach

This study applies novel risk measures, called Orlicz risks, to the risk and uncertainty evaluation of the streamflow discharge as a primary driver of hydrological and hydraulic processes of interest in civil and environmental engineering. We consider the mixed moving average process governing the discharge whose statistics are explicitly represented as some product of a time-scale characterizing the flow attenuation and a jump moment governing the size and frequency of jumps. The classical Orlicz risks are extended so that not only the upper tail risk but also the lower one of the jump size and attenuation of the discharge can be evaluated within a single mathematical framework. Further, the risk and uncertainty can be individually quantified in a tractable manner by the proposed Orlicz risks. Computing the Orlicz risks reduces to solving a pair of novel static optimization problems that are solvable semi-analytically. The risk and uncertainty involved in the streamflow dynamics can be consistently evaluated by specifying few user-dependent parameters. The associated Radon–Nikodym derivatives as the worst-case model uncertainties are obtained as byproducts. Sufficient conditions for the well-posedness of the Orlicz risks are discussed and numerical algorithms for computing them are presented. We finally apply the proposed framework to a statistical analysis of the streamflow discharge time series data collected at mountainous river environments.

Application of clustering cooperative differential privacy in spatial crowdsourcing task allocation

A framework for spatial crowdsourcing task allocation based on centralized differential privacy is proposed for addressing the problem of worker’s location privacy leakage. Firstly, by combining two stages of differential privacy noise addition and clustering matching, a spatial crowdsourcing worker dataset with high differential privacy protection can be obtained; Secondly, the dynamic problem of spatial crowdsourcing task allocation is transformed into a static combinatorial optimization problem by dividing the spatiotemporal units and the “delay matching” strategy; Finally, the improved discrete glowworm swarm optimization algorithm is used to calculate the results of spatial crowdsourcing task allocation. It has been demonstrated that, compared to the direct differential privacy noise-adding assignment method and the discrete glowworm swarm optimization assignment method, the proposed method achieves better task assignment results, with the total travel distance reduced by 12.42% and 3.56%, respectively, and the task assignment success rate increased by 11.75% and 3.34%, respectively.

Dynamic In Vivo Computation for Learning-Based Nanobiosensing in Time-Varying Biological Landscapes

We have recently proposed a framework of in vivo computation which transforms the early tumor sensing problem into a computational problem. In the framework, a tumor-triggered biological gradient field (BGF) guides swarm-intelligence-assisted targeting process, where externally manipulable and trackable magnetic nanorobots act as computational agents for the optimization procedure. As BGF can be viewed as an objective function which is utilized to define the fitness landscape for the agents, the inherent attributes of BGF are critical to the in vivo computational process. All our previous investigations are based on the hypothesis that the BGF landscape remains time-invariant during the tumor targeting process, which results in a static function optimization problem. However, the properties of internal environment, such as the flow state of body fluid, will naturally lead to time-dependent variation of BGF, which means that the targeting process should be modeled as a dynamic function optimization problem. Based on this consideration, we focus on dynamic in vivo computation by considering different variation patterns of BGF in this paper. Two computational intelligence strategies named “swarm-based learning” and “individual-based learning” are proposed for dealing with the turbulence of the fitness estimation caused by the BGF variation. The in silico experiments and statistical results demonstrate the effectiveness of the proposed strategies. In addition, the above process is conducted in a three-dimensional search space, where the tumor vascular network is generated by an invasion percolation algorithm, which is more realistic compared to the two-dimensional search space in our previous works.

ON THE TREATMENT OF REQUIREMENTS IN DFAM: THREE INDUSTRIAL USE CASES

AbstractOptimization-driven design offers advantages over traditional experience-based mechanical design. As an example, topology optimization can be a powerful tool to generate body shapes for Additive Manufacturing (AM). This is helpful, when (1) load paths are non-intuitive due to complex design domains or boundary conditions, or (2) the design process is to be automated to minimize effort associated with experience-based design. However, practically relevant boundary conditions are often difficult to put into a formal mathematical language to, for example, either feed it into a topology optimization algorithm, or provide precise quantitative criteria for CAE-supported manual design. This paper presents a survey of three industry use cases and identifies three types of requirements: the first can be directly cast into parts of an optimization problem statement (∼ 40%), the second is considered indirectly by adapting the optimization problem without explicit reference to the requirement (∼ 20%), and the third is only assessed after the design is finalized (∼ 40%). For categories 2 and 3 we propose directions of improvement to support formulating complex design tasks as unambiguous design problems.

Focus programming: a bi‐level programming approach to static stochastic optimization problems

AbstractStatic stochastic optimization problems are formulated with the focus theory of choice where the optimal solution is determined as per which solution's focus (the most salient realization of a random vector) is the most preferred. The new formulation that we call the focus programming is a bi‐level programming approach in which the lower‐level program is used to choose the focus of each feasible solution and the upper‐level program is to determine the optimal solution. Since in focus programming models upper‐level and lower‐level programs are maximin or minimax problems, they are nonsmooth and sometimes even nonconvex so that the existing optimization methods cannot solve such bi‐level programming problems. We propose several single‐level reformulation methods for such problems.

Dynamic Optimization of Role Concepts for Role-Based Access Control Using Evolutionary Algorithms

To ensure the security of current information systems, role-based access control (RBAC) is a widely used concept. For this purpose, based on an initial assignment of permissions to users, permissions are grouped to roles, which are then assigned to users. The corresponding (NP-complete) optimization problem, the so-called role mining problem (RMP), aims at finding a minimal set of roles and a corresponding assignment of those roles to users. Previously, the RMP has been considered as a static optimization problem. However, the application of RBAC in real business use cases requires the inclusion of dynamically occurring events that reflect changes in the business environment of companies, as well as events that result from direct user interaction with the role mining process. Therefore, in this paper, we provide a comprehensive overview and classification of the most relevant events for role mining and present methods for integrating them into the framework of an evolutionary role mining algorithm. The functional performance of these methods as well as the overall performance gain of dynamic role mining compared to the static approach is then examined in a series of experiments.

Optimal proportional and excess-of-loss reinsurance for multiple classes of insurance business

In this paper we consider a reinsurance strategy which combines a proportional and an excess-of-loss reinsurance in a risk model with multiple dependent classes of insurance business. Under the assumption that the claim number of the classes has a multivariate Poisson distribution, the aim is to maximize the expected utility of terminal wealth. In a general setting, after deriving the corresponding Hamilton–Jacobi–Bellman equation, we prove a Verification Theorem and identify sufficient conditions for the optimality. Then, in a special case with exponential utility, an explicit solution is found by solving an intricate associated static constrained optimization problem.

Application of metaheuristic optimization based support vector machine for milling cutter health monitoring

With the arrival of Industry 4.0, intelligent condition-based maintenance has become a must, if not a need, for industries with significant capital investments in rotating machineries. Tool Condition Monitoring (TCM) is one of the strategic research domains in condition-based maintenance. Lately, supervised algorithms based on Machine Learning (ML) techniques assist classification of the cutting tool's condition in operation. One such algorithm is the Support Vector Machine (SVM) popularly used for training the data however, choosing optimal hyper-parameters for an SVM is essential in making the model robust. Owing to intermittent cutting in a milling operation, the modeling of tool conditions based on vibrations evolved during machining needs to be handled wisely. Consequently, there exists a need for meta-heuristic optimization algorithms to drive SVM for evaluating the robustness of the model and to increase accuracy, thereby minimizing the risk of false classification of tool bits. Over the past decade, meta-heuristic algorithms have found immense use in optimizing ML models and solving real–life engineering problems. This research paper aims to optimize hyperparameters of SVM – ‘C’ and ‘gamma’ using metaheuristic algorithms in the context of TCM. Further, the paper evaluates popular metaheuristic algorithms. It compares their respective efficacies, enabling researchers in the field of TCM to choose the appropriate algorithm for their optimization problem statement to get higher performance predictions from their SVM models.

Mean Field Markov Decision Processes

We consider mean-field control problems in discrete time with discounted reward, infinite time horizon and compact state and action space. The existence of optimal policies is shown and the limiting mean-field problem is derived when the number of individuals tends to infinity. Moreover, we consider the average reward problem and show that the optimal policy in this mean-field limit is varepsilon -optimal for the discounted problem if the number of individuals is large and the discount factor close to one. This result is very helpful, because it turns out that in the special case when the reward does only depend on the distribution of the individuals, we obtain a very interesting subclass of problems where an average reward optimal policy can be obtained by first computing an optimal measure from a static optimization problem and then achieving it with Markov Chain Monte Carlo methods. We give two applications: Avoiding congestion an a graph and optimal positioning on a market place which we solve explicitly.

Static and dynamic topology optimization: an innovative unifying approach

This paper presents a topology optimization approach that is innovative with respect to two distinct matters. First of all the proposed formulation is capable to handle static and dynamic topology optimization with virtually no modifications. Secondly, the approach is inherently a multi-input multi-output one, i.e., multiple objectives can be pursued in the presence of multiple loads. The input-to-output transfer matrix, say {{{varvec{G}}}}, is the key ingredient that governs the algebraic mapping between applied loads and structural response. In statics {{{varvec{G}}}} depends on the design variables only, whereas it depends on the frequency variable as well in the dynamic case. The Singular Value Decomposition (SVD) of {{{varvec{G}}}} represents then the core of the proposed approach. Singular values are shown to be the gains of the input/output mapping and are used to compute proper norms of {{varvec{G}}} that represent the goal functions to be minimized. Singular vectors provide at no extra cost the plant directions, i.e., the load combination factors that stress the structure the most. Numerical examples are discussed in much detail and open issues object of ongoing investigations are highlighted. A full Matlab code handling the static topology optimization problem is provided as an online Appendix to the manuscript. Its extension to the dynamic case may be gathered following the formulation proposed in Sect. 5.

Формирование структуры интерфейса: оптимальные задачи

Discusses the problems that arise in the process of interface design at the initial design stage. At this stage, the main necessary system functions and user scenarios are determined; additionally, a general interface structure is formed, intended for a generalized representation of the future interface, highlighting the main and secondary areas, objects, functions, which identifies the type and direction of connecting components and blocks, as well as a sequence of commands and actions. As a model, the design of the general interface structure is considered as the definition of a number (set) of rules that determine the importance and sequence of using the existing set of functional blocks, which are then treated as vertices of a digraph, while the need to provide (possibility) transitions from one block to another is treated as transitions between vertices of a digraph (arc). A thorough discussion is provided describing in detail the process of constructing a basic graph and determining the necessary parameters of vertices and arcs, based on the integrated use of the relevant experts’ knowledge (users and developers). A number of possible statements of unconstrained optimization problems on the resulting graph and their features are formulated and analyzed. The problems of conditional optimization on the obtained graph are defined and studied. The proposed approach and the solution of optimization problems at the appropriate stages will help to determine the most efficient structure of the interface designed from the point of view of the user and system capabilities, as well as under limited time and other resources to complete tasks.

A real-time balancing market optimization with personalized prices: From bilevel to convex

This paper studies the static economic optimization problem of a system with a single aggregator and multiple prosumers in a Real-Time Balancing Market (RTBM). The aggregator, as the agent responsible for portfolio balancing, needs to minimize the cost for imbalance satisfaction in real-time by proposing a set of optimal personalized prices to the prosumers. On the other hand, the prosumers, as price taker and self-interested agents, want to maximize their profit by changing their supplies or demands and providing flexibility based on the proposed personalized prices. We model this problem as a bilevel optimization problem. We first show that the optimal solution of this bilevel optimization problem can be found by solving an equivalent convex problem. In contrast to the state-of-the-art Mixed-Integer Programming (MIP)-based approach to solve bilevel problems, this convex equivalent has very low computation time and is appropriate for real-time applications. Next, we compare the optimal solutions of the proposed personalized scheme and a uniform pricing scheme. We prove that, under the personalized pricing scheme, more prosumers contribute to the RTBM and the aggregator’s cost is less. Finally, we verify the analytical results of this work by means of numerical case studies and simulations.

An exact algorithm for the static pricing problem under discrete mixed logit demand

Price differentiation is a common strategy in many markets. In this paper, we study a static multiproduct price optimization problem with demand given by a discrete mixed multinomial logit model. By considering a mixed logit model that includes customer specific variables and parameters in the utility specification, our pricing problem reflects well the discrete choice models used in practice. To solve this pricing problem, we design an efficient iterative optimization algorithm that asymptotically converges to the optimal solution. To this end, a linear optimization (LO) problem is formulated, based on the trust-region approach, to find a “good” feasible solution and approximate the problem from below. A convex optimization problem is designed using a convexification technique to approximate the optimization problem from above. Then, using a branching method, we tighten the optimality gap. The effectiveness of our algorithm is illustrated on several cases, and compared against solvers and existing state-of-the-art methods in the literature.