Lower Bound Approach Research Articles

Optimal drift rate control and two-sided impulse control for a Brownian system with the long-run average criterion

Abstract In this paper, we consider a joint drift rate control and two-sided impulse control problem in which the system manager adjusts the drift rate as well as the instantaneous relocation for a Brownian motion, with the objective of minimizing the total average state-related cost and control cost. The system state can be negative. Assuming that instantaneous upward and downward relocations take a different cost structure, which consists of both a setup cost and a variable cost, we prove that the optimal control policy takes an $\left\{ {\!\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right),\!\left\{ {{\mu ^{\ast}}(x)\,:\,x \in [ {{s^{\ast}},{S^{\ast}}}]} \right\}} \right\}$ form. Specifically, the optimal impulse control policy is characterized by a quadruple $\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right)$ , under which the system state will be immediately relocated upwardly to ${q^{\ast}}$ once it drops to ${s^{\ast}}$ and be immediately relocated downwardly to ${Q^{\ast}}$ once it rises to ${S^{\ast}}$ ; the optimal drift rate control policy will depend solely on the current system state, which is characterized by a function ${\mu ^{\ast}}\!\left( \cdot \right)$ for the system state staying in $[ {{s^{\ast}},{S^{\ast}}}]$ . By analyzing an associated free boundary problem consisting of an ordinary differential equation and several free boundary conditions, we obtain these optimal policy parameters and show the optimality of the proposed policy using a lower-bound approach. Finally, we investigate the effect of the system parameters on the optimal policy parameters as well as the optimal system’s long-run average cost numerically.

Dietary exposure to polyaromatic hydrocarbons of the Hong Kong population

Polyaromatic hydrocarbons (PAHs) are ubiquitous in the environment and food. The Joint FAO/WHO Expert Committee on Food Additives concluded 13 individual PAHs are carcinogenic and genotoxic in vitro and in vivo. Food is recognized as the main source of exposure to PAHs for adult non-smokers, which contributed to more than 90% of total exposure. In this study, 300 food samples were collected in Hong Kong, analysed the levels of 16 European Union priority PAHs, the dietary exposure to these PAHs by the local adult population from these food items, and the associated health risk. The most predominant detectable PAH was chrysene (CHR) (14.4%), followed by benzo[c]fluorene (11.2%), benzo[a]anthracene (BaA) (10.6%) and benzo[b]fluoranthene (BbFA) (7.8%). The dietary exposures for average consumers of benzo[a]pyrene (BaP) and PAH4 (sum of BaP, CHR, BaA and BbFA) were 0.13–0.90 and 1.4–4.2 ng/kg bw/day respectively for lower and upper bound approaches. Cereal and its products contributed more than 50% to BaP and PAH4 for average consumers in a lower-bound approach. The margin of exposure (MOE) approach was used to assess the health risks of consumers. The calculated MOE values for both BaP and PAH4 of the average and high consumers (90th percentile) were >50,000, indicating a low concern for the health of the Hong Kong population.

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

Asymptotically Optimal Control of Make-to-Stock Systems

We consider multiclass make-to-stock production/inventory systems in which the manager makes three decisions, including pricing, outsourcing, and scheduling, to maximize the long-run average profit. For a sequence of systems in the heavy-traffic regime, with linear or strictly convex holding/waiting cost functions, we propose a sequence of policies and establish its asymptotic optimality. Our proof combines the lower bound approach and a thorough steady-state analysis of the systems. We also establish general results on the existence and tightness of the stationary distributions of the state processes under a more general family of policies. Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819 and 14505820] and the National Natural Science Foundation of China [Grant 72222023].

Dexterous workspace optimization for a six degree-of-freedom parallel manipulator based on surrogate-assisted constrained differential evolution

Parallel manipulators are increasingly applied because of their high stiffness, load capacity, and motion precision. However, the workspace of a parallel manipulator is relatively small compared to that of a serial manipulator. Therefore, appropriate structural designs of parallel manipulators for maximizing the workspace are essential. This paper studies the dexterous workspace maximization problem of a six-degree-of-freedom (DOF) parallel manipulator. First, a constrained optimization model is formulated, in which, the objective is to maximize the dexterous workspace, and the constraints include several kinematic and dynamic indicators, and some general structural limitations. To address the highly constrained, computationally-expensive problem, a surrogate-assisted two-phase constrained differential evolution method is proposed. Its effectiveness and efficiency are verified by the superior performance over two surrogate-free evolutionary methods: differential evolution (DE) and genetic algorithm (GA), and two surrogate-assisted state-of-the-art evolutionary methods: modified constrained expected improvement (CEI) and the lower confidence bounding approach (PCLCB), on ten benchmark functions and the established real-word 6-DOF manipulator design optimization problem.

Radio-[formula omitted]-labeling of cycles for large [formula omitted

Let G be a simple connected graph. For any two vertices u and v, let d(u,v) denote the distance between u and v in G. A radio-k-labeling of G for a fixed positive integer k is a function f which assigns to each vertex a non-negative integer label such that for every two vertices u and v in G, |f(u)−f(v)|⩾k−d(u,v)+1. The span of f is the difference between the largest and smallest labels of f(V). The radio-k-number of a graph G, denoted by rnk(G), is the smallest span among all radio-k-labelings admitted by G. A cycle Cn has diameter d=⌊n/2⌋. In this paper, we combine a lower bound approach with cyclic group structure to determine the value of rnk(Cn) for k⩾n−3. For d⩽k<n−3, we obtain the values of rnk(Cn) when n and k have the same parity, and prove partial results when n and k have different parities. Our results extend the known values of rnd(Cn) and rnd+1(Cn) shown by Liu and Zhu (2005), and by Karst et al. (2017), respectively.

A three-dimensional approach to the Extended Limit Analysis of Reinforced Masonry

The Extended Limit Analysis of Reinforced Masonry (ELARM) is a simple and user-friendly method for the design and structural analysis of singly-curved, reinforced tile vaults [1]. It is based on limit analysis but takes into account the reinforcement’s contribution to the composite cross-section’s bending capacity.A three-dimensional approach to ELARM is presented in this paper. The theoretical framework to understand the implications and limitations of extending ELARM to fully 3D structures is described, together with the strategies to carry out the leap from 2D to 3D. This extension is a lower-bound approach for the design of reinforced masonry, reinforced concrete and concrete-masonry composite shells and the assessment of their strength and stability against external loading.The new, extended method is implemented computationally to speed up the iterative processes, provide quick structural feedback, offer immediate results and allow for user-interactive form-finding and optimisation procedures. Different applications of the developed tool are described through the presentation of examples, including reinforcement optimisation, a form-finding process and a case with a shape beyond funicular geometry.

Optimization design of antenna cover of deeply submerged body based onparallel variable-fidelity LCB algorithm

Objectives In order to reduce the weight while maintaining the performance of the antenna cover of a deeply submerged body, a parallel variable-fidelity lower confidence bound (PVF-LCB) approach is proposed to optimize the structure of the antenna cover. Methods The proposed algorithm adaptively allocates computational resources of different fidelities through a variable-fidelity LCB (VF-LCB) function, allowing it to select several candidate samples based on influence functions (IFs) constructed by the variable-fidelity Kriging model. Moreover, the proposed method is assisted with widely used constraint-handling methods to solve the structural optimization problem. Results The proposed approach obtains an optimized antenna cover structure which satisfies all constraints. Compared with the well-known variable-fidelity optimization method, the optimized structure is about 50% lower in weight. Additionally, the proposed approach reduces the weight of the antenna cover by about 30% compared with the results of single-fidelity parallel optimization methods. Conclusions The proposed method can not only reduce the design cycle of engineering optimization, but improve the quality of the optimal solution, giving it certain development prospects and guiding significance for engineering applications.

Alexander Revisited: Upper- and Lower-Bound Approaches for Axial Crushing of a Circular Tube

In the classical Alexander model for a rigid, perfectly plastic circular tube under axial progressive crushing, an expression for the average crushing force over one complete folding cycle was given in terms of the folding length while the value of the folding length was determined by minimising the average force. This approach is often misunderstood as a correct upper-bound method and has been widely used for analysing large plastic deformation of structures. In this paper, it is highlighted that Alexander approach is strictly not an upper-bound analysis and alternative analyses are presented. A new initial collapse mechanism involving folding both inwards and outwards of the tube wall is proposed for an upper-bound analysis, which leads to the determination of the folding characteristics such as the folding length. Subsequently, a repeatable mechanism is idealised for the periodic folding process. The upper-bound analysis is performed by minimizing the instant force required for maintaining the folding process, which is equivalent to adopting the energy balance for the incremental plastic deformation rather than the total deformation of one complete folding cycle. Instead of merely obtaining the average force as Alexander did, the present theoretical analysis gives a complete force-displacement curve as well as plastic folding length, etc., which agrees well with the existing knowledge. Furthermore, an equilibrium approach is presented involving detail stress state for a deforming configuration of the tube. The lower-bound results re-produce the variation of the force as obtained from the upper-bound analysis when the folding configurations are the same as those in the assumed collapse mechanisms. The present study conceptually reconciles the upper-bound and low-bound theories in structural plasticity for large deformation analysis.

Ergodic inventory control with diffusion demand and general ordering costs

In this work, we consider a continuous-time inventory system where the demand process follows an inventory-dependent diffusion process. The ordering cost of each order depends on the order quantity and is given by a general function, which is not even necessarily continuous and monotone. By applying a lower bound approach together with a comparison theorem, we show the global optimality of an (s,S) policy for this ergodic inventory control problem.

Sequencing mixed-model assembly lines with risk-averse stochastic mixed-integer programming

Sequencing decisions in mixed-model assembly lines are complicated by various uncertainty factors. This paper addresses a real-life uncertainty factor identified in a manufacturer of large vehicles, by modelling unreliable part delivery and quality. Stochastic optimisation is applied to find sequencing policies that improve the on-time performance of its mixed-model assembly lines. As schedulers have different levels of risk aversion, a risk-averse programme is further presented to protect against the decision maker’s chosen fraction of worst scenarios. Computational studies with Progressive Hedging as the solution method, and its lower bounding approach, demonstrate the high quality of resulting sequencing decisions and the time efficiency of the solution method.

Perfluoroalkyl contaminants in eggs from backyard chickens reared in Italy

Per- and poly-fluoroalkyl substances (PFASs) are persistent and bioaccumulative compounds with adverse impacts on the environment and human health. Diet is one of the main sources of exposure to PFASs. Recently, the EFSA established a tolerable weekly intake (TWI) limit (4.4 ng/kg b.w.) for a mixture of the four major PFASs. Eggs and egg products can contribute to this intake, with their contamination possibly dependent on the husbandry system. Monitoring Italian eggs from backyard chickens revealed a relatively uniform PFAS contamination, with perfluoro-1-octanesulfonate being the most abundant. Contamination was detected to be significantly higher in eggs from backyard chickens than in eggs from commercial laying hens, consistent with a previous Italian study. According to the recently set TWI value, the consumption of eggs from backyard chickens could contribute significantly to dietary intake of PFASs (up to 29% of the TWI in children, considering the lower bound approach).

Quadrature spatial modulation‐aided single‐input multiple‐output‐media‐based modulation

SummarySingle‐input multiple‐output media‐based modulation (SIMO‐MBM) improves the limitation of single‐input multiple‐output (SIMO) systems by minimizing the required number of antennas to achieve a high data rate and enhanced error performance. In this paper, we employ the quadrature dimension of the spatial constellation by decomposing the amplitude/phase modulation (APM) symbol into real and imaginary components similar to quadrature spatial modulation (QSM) to improve the overall error performance of the conventional SIMO‐MBM and to achieve a higher data rate. The average bit error probability of the proposed scheme is formulated using a lower bound approach and validated by the Monte Carlo (MC) simulation results obtained. Furthermore, the effect of the suboptimal mirror activation pattern selection using channel amplitude coupled with antenna correlatio n is examined for the proposed system. The MC simulation results achieved demonstrate a substantial improvement of 3.5 dB at 10 b/s/Hz with mrf = 2 and 7 dB at 12 b/s/Hz with mrf = 4 over the conventional SIMO‐MBM scheme.

A parallel constrained lower confidence bounding approach for computationally expensive constrained optimization problems

Computationally expensive constrained optimization problems (COPs) are very common in real-world engineering designs. This work proposes a parallel constrained lower confidence bounding (PCLCB) approach to solve challenging optimization problems with computationally expensive objective and constraints. In the PCLCB approach, a two-phase constrained lower confidence bounding (CLCB) criterion is proposed to adaptively allocate infill sample points. In the first phase where no feasible solution exists, a lower confidence bounding (LCB) function of constraints is developed to search for feasible points and improve the accuracies of the Kriging models. After a feasible point is found, the CLCB criterion focuses on local search in the second phase to improve the obtained optimum. In this phase, the search is implemented in a limited region where points may be feasible. Furthermore, an enhanced Kriging believer strategy is presented to parallelize the CLCB criterion to generate a batch of candidate points at each iteration. The PCLCB approach is tested on sixteen numerical benchmark cases and compared with the state-of-the-art alternative methods. Results indicate that the performance of PCLCB is highly competitive to the recently published algorithms in terms of efficiency and effectiveness. To verify the capability of PCLCB in solving real-world engineering problems, the lightweight design of micro-aerial vehicle (MAV) fuselage is presented. The weight of the MAV fuselage obtained by the proposed approach is on average 3.5% less than that of the other alternative methods.

Impulse Control with Discontinuous Setup Costs: Discounted Cost Criterion

This paper studies a continuous-review backlogged inventory model considered by [K. L. Helmes, R. H. Stockbridge, and C. Zhu, SIAM J. Control. Optim., 53 (2015), pp. 2100--2140] but with discontinuous quantity-dependent setup cost for each order. In particular, the setup cost is characterized by a two-step function and a higher cost would be charged once the order quantity exceeds a threshold $Q$. Unlike the optimality of the $(s,S)$-type policy obtained by Helmes, Stockbridge, and Zhu for continuous setup cost with the discounted cost criterion, we find that, in our model, although some $(s,S)$-type policy is indeed optimal in some cases, the $(s,S)$-type policy cannot always be optimal. In particular, we show that there exist cases in which an $(s,S)$ policy is optimal for some initial levels but it is strictly worse than a generalized $(s,\{S(x):x\leq s\})$ policy for the other initial levels. Under $(s,\{S(x):x\leq s\})$ policy, it orders nothing for $x>s$ and orders up to level $S(x)$ for $x\leq s$, where $S(x)$ is a nonconstant function of $x$. We further prove the optimality of such $(s,\{S(x):x\leq s\})$ policy in a large subset of admissible policies for those initial levels. Moreover, the optimality is obtained through establishing a more general lower bound theorem which will also be applicable in solving some other optimization problems by the lower bound approach.

Ordering for Communication-Efficient Quickest Change Detection in a Decomposable Graphical Model

A quickest change detection problem is considered in a sensor network with observations whose statistical dependency structure across the sensors before and after the change is described by a decomposable graphical model (DGM). Distributed computation methods for this problem are proposed that are capable of producing the optimum centralized test statistic. The DGM leads to the proper way to collect nodes into local groups equivalent to cliques in the graph, such that a clique statistic which summarizes all the clique sensor data can be computed within each clique. The clique statistics are transmitted to a decision maker to produce the optimum centralized test statistic. In order to further improve communication efficiency, an ordered transmission approach is proposed where transmissions of the clique statistics to the fusion center are ordered and then adaptively halted when sufficient information is accumulated. This procedure is always guaranteed to provide the optimal change detection performance, despite not transmitting all the statistics from all the cliques. A lower bound on the average number of transmissions saved by ordered transmissions is provided and for the case where the change seldom occurs the lower bound approaches approximately half the number of cliques provided a well behaved distance measure between the distributions of the sensor observations before and after the change is sufficiently large. We also extend the approach to the case when the graph structure is different under each hypothesis. Numerical results show significant savings using the ordered transmission approach and validate the theoretical findings.

An Entropy Weight-Based Lower Confidence Bounding Optimization Approach for Engineering Product Design

The optimization design of engineering products involving computationally expensive simulation is usually a time-consuming or even prohibitive process. As a promising way to relieve computational burden, adaptive Kriging-based design optimization (AKBDO) methods have been widely adopted due to their excellent ability for global optimization under limited computational resource. In this paper, an entropy weight-based lower confidence bounding approach (EW-LCB) is developed to objectively make a trade-off between the global exploration and the local exploitation in the adaptive optimization process. In EW-LCB, entropy theory is used to measure the degree of the variation of the predicted value and variance of the Kriging model, respectively. Then, an entropy weight function is proposed to allocate the weights of exploration and exploitation objectively and adaptively based on the values of information entropy. Besides, an index factor is defined to avoid the sequential process falling into the local regions, which is associated with the frequencies of the current optimal solution. To demonstrate the effectiveness of the proposed EW- LCB method, several numerical examples with different dimensions and complexities and the lightweight optimization design problem of an underwater vehicle base are utilized. Results show that the proposed approach is competitive compared with state-of-the-art AKBDO methods considering accuracy, efficiency, and robustness.

Ground-state correlation energy of counterions at a charged planar wall: Gibbs-Bogoliubov lower-bound approach

Recent simulation results imply the lowering of the ground-state correlation energy per counterion at a charged planar wall, compared with that of the 2D and 3D one-component plasma systems. Our aim is to correctly evaluate the ground-state energy of strongly-coupled counterion systems by considering a quasi-2D bound state where bound counterions are confined to a layer of molecular thickness. We use a variational approach based on the Gibbs-Bogoliubov inequality for the lower-bound free energy so that the liquid-state theory can be incorporated into the formulations. The soft mean spherical approximation demonstrates that the lowered ground-state energy can be reproduced by the obtained analytical form of a quasi-2D bound state.

Design method for quantifying embankment safety against lateral spreading and determining contribution of basal reinforcements

This study proposes an analytical method for the calculation of the safety factor against lateral spreading in embankments, both for cases with and without basal reinforcement. Hence, a new limit equilibrium approach is presented based on the lower bound plasticity theorem, which allows the computation of the required distribution of stresses along presumed slip surfaces at the limit state. Comparing the required distribution of stresses with the available resistances, the safety factor in the absence of basal reinforcement is calculated. If the safety factor is unsatisfactory, basal reinforcement can be introduced and its contribution to stability quantified. The lower bound approach allows the computation of mobilized resistance along the reinforcement. Accordingly, the proposed method allows the calculation of the safety factor against lateral spreading for problems involving basal reinforcements. Finally, the proposed method is compared with well-established methods in literature. Results suggest that the proposed method is reliable, practical, and applicable to safe and economical designs.

Variable-Fidelity Lower Confidence Bounding Approach for Engineering Optimization Problems with Expensive Simulations

Engineering design optimization with expensive simulations is usually a computationally prohibitive process. As one of the most famous efficient global optimization approaches, the lower confidence bounding (LCB) approach has been widely applied to relieve this computational burden. However, the LCB approach can be used only for design optimization problems with the single-fidelity level. In this paper, a variable-fidelity lower confidence bounding (VF-LCB) approach is developed to extend the LCB approach to engineering problems with multifidelity levels. First, a VF-LCB function is analytically derived to adaptively select LF or HF samples according to the predicted values and uncertainty from the VF metamodel. Then the coefficient of variations (CoV) of the predicted values and uncertainty of the VF model, which can reflect the degree of dispersion of the prediction values and uncertainty, respectively, are introduced to balance the global exploration and local exploitation objectively. To illustrate the effectiveness and merits of the proposed VF-LCB approach, eight numerical examples, and the design of micro-aerial vehicle (MAV) fuselage are tested. Comparative results between the proposed approach and the other five existing methods show that the average cost savings are about 25% for eight numerical examples and about 45% for the MAV problem compared with the other five existing methods.