Lower-bound Constraints Research Articles

On the existence of a solution of one variational inequality of the nonlinear filtration theory

A theorem on the existence of a generalized solution of one variational inequality describing the process of nonlinear nonstationary filtration of a liquid in a porous medium with the condition of one-way permeability on a part of the boundary is proved. The case is considered, in which the Kirchhoff transformation used in the determination of a generalized solution maps the real axis to the semi-axis bounded from below. In investigating the solvability of the resulting variational inequality with a lower-bound constraint on the solution, an auxiliary problem with no constraints is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem studied in the paper. The solvability of the auxiliary problem is established by means of using the semidiscretization method with a penalty and the Galerkin method.

Constrained evolutionary games by using a mixture of imitation dynamics

Game dynamics have been widely used as learning and computational tool to find evolutionarily stable strategies. Nevertheless, most of the existing evolutionary game dynamics, i.e., the replicator, Smith, projection, Brown–Von Neumann–Nash, Logit and best response dynamics have been analyzed only in the unconstrained case. In this work, we introduce novel evolutionary game dynamics inspired from a combination of imitation dynamics. The proposed approach is able to satisfy both upper- and lower-bound constraints. Moreover, dynamics have asymptotic convergence guarantees to a generalized-evolutionarily stable strategy. We show important features of the proposed game dynamics such as the positive correlation and invariance of the feasible region. Several illustrative examples handling population state constraints are provided.

Reliability enhancement of mixed-domain seismic inversion with bounding constraints

ABSTRACTSeismic inversion works as one of pragmatic and effective approaches to estimate the subsurface parameters. One robust Bayesian sparse inversion method incorporating the mixed-domain convolution with model bounding constraints is proposed in this study. The time-domain response and partial frequency components are utilized in mixed-domain seismic inversion to improve the resolution and stability of seismic inversion. First, the objective updated function is yielded with Bayesian inference in joint time and frequency domain. And, the sparse constraint is incorporated into the objective function to improve robustness of inversion algorithm. Conventional seismic inversion methods always don’t focus on the lower and upper bounding constraints on subsurface model parameters, due to which unrealistic predicted results may arise. To get rid of the problem, the bounding constraints on P-wave impedance elaborated by logarithmic or inverse hyperbolic tangent formulas is introduced in mixed-domain seismic inversion as it renders to reduce the unrealistic parameters and enhance the reliability of prediction effectively. In addition, the synthetic examples demonstrate the effectiveness and robustness of the proposed inversion algorithm. Finally, one field case is studied carefully and the estimated parameters with bounding constraints at the borehole-side location can preserve a high degree of agreements with real logging data to verify the practicability of the bounding-constraining mixed-domain Bayesian inversion.

Hybrid Optimal Control under Mode Switching Constraints with Applications to Pesticide Scheduling

This paper concerns optimal mode-scheduling in autonomous switched-mode hybrid dynamical systems, where the objective is to minimize a cost-performance functional defined on the state trajectory as a function of the schedule of modes. The controlled variable, namely the modes’ schedule, consists of the sequence of modes and the switchover times between them. We propose a gradient-descent algorithm that adjusts a given mode-schedule by changing multiple modes over time-sets of positive Lebesgue measures, thereby avoiding the inefficiencies inherent in existing techniques that change the modes one at a time. The algorithm is based on steepest descent with Armijo step sizes along Gâteaux differentials of the performance functional with respect to schedule-variations, which yields effective descent at each iteration. Since the space of mode-schedules is infinite dimensional and incomplete, the algorithm’s convergence is proved in the sense of Polak’s framework of optimality functions and minimizing sequences. Simulation results are presented, and possible extensions to problems with dwell-time lower-bound constraints are discussed.

Patron Allocation for Group Services Under Lower Bound Constraints

Group services are highly important for a variety of computing application domains. In this paper, we study the fundamental problem of allocating a set of service patrons to a set of service groups in an attempt to maximize the total profit gained by the grouping platform. The problem under consideration is unique in that group service is not provided at all unless its lower bound requirement is satisfied. In addition, we allow each service patron to join multiple groups. In this paper, after proving the hardness property of the problem, we focus first on a special case of the problem. To this end, we propose two approaches. One aims at providing a suboptimal solution using a 1/2-approximation algorithm. The other approach turns to seeking an optimal solution using a branch and bound technique. For this purpose, we introduce a theorem that captures a useful property of an optimal allocation. Based on this theorem, we design an efficient branch and bound algorithm to find an optimal solution. We then extend these methods to solve the general problem. Extensive experiments show that our branch and bound algorithm is able to obtain an optimal solution with a small amount of computation time in many different settings.

Advantages of an Ellipse when Modeling Leisure Utility

This paper characterizes a specification for the utility of leisure that is based on the general equation for an ellipse. We show that this functional form has multiple benefits. The elliptical utility function provides Inada conditions at both the upper-bound and lower-bound constraints on labor supply, which is not the case for the two most common alternative functions. The presence of these two Inada conditions in the elliptical utility of leisure specification speeds up the computation by a factor between three and six times. We further show that the elliptical utility of leisure function is a close approximation to the constant relative risk aversion and constant Frisch elasticity functions in terms of marginal utilities, microeconomic outcomes in a life cycle model, and macroeconomic outcomes in a simple real business cycle model.

Optimum design of thermally loaded beam-columns for maximum vibration frequency or buckling temperature

With the cross-sectional area function as the design variable, we seek the optimum design of clamped–clamped, thin, linearly elastic beam-columns under thermal load that maximizes the buckling temperature or the fundamental natural frequency of transverse vibrations. The beam-columns have given length, geometrically similar cross-sections of variable size and a given volume of material. A lower and/or upper bound constraint may be prescribed for the cross-sectional area of the beam-columns. To account for possible bimodality of the optimum designs, both the unimodal and bimodal optimality criterion approaches are applied. For a lower bound smaller than a specific value, bimodal optimum designs are obtained for problems of maximum buckling temperature. When maximizing the thermal fundamental natural frequency, optimum designs may be uni- or bimodal depending on the values of the given thermal load and the lower bound on the cross-sectional area. The geometrically unconstrained optimum design for maximum fundamental natural frequency obtained by Olhoff (1976) without consideration of thermal load, is shown to be at the same time an optimum design that maximizes the fundamental natural frequency at any temperature rise, but it is found to be associated with zero buckling load. Moreover, the optimum design maximizing the critical buckling temperature is very similar to that maximizing the mechanical buckling load obtained by Olhoff and Rasmussen (1977). This interesting analogy is explained by studying optimum design for minimum axial compressive force.

Switched-mode systems: gradient-descent algorithms with Armijo step sizes

This paper concerns optimal mode-scheduling in autonomous switched-mode hybrid dynamical systems, where the objective is to minimize a cost-performance functional defined on the state trajectory as a function of the schedule of modes. The controlled variable, namely the modes’ schedule, consists of the sequence of modes and the switchover times between them. We propose a gradient-descent algorithm that adjusts a given mode-schedule by changing multiple modes over time-sets of positive Lebesgue measures, thereby avoiding the inefficiencies inherent in existing techniques that change the modes one at a time. The algorithm is based on steepest descent with Armijo step sizes along Gâteaux differentials of the performance functional with respect to schedule-variations, which yields effective descent at each iteration. Since the space of mode-schedules is infinite dimensional and incomplete, the algorithm’s convergence is proved in the sense of Polak’s framework of optimality functions and minimizing sequences. Simulation results are presented, and possible extensions to problems with dwell-time lower-bound constraints are discussed.

Linear programming approach to design of spatial link mechanism with partially rigid joints

A simple systematic approach is presented for designing a spatial link mechanism with partially rigid joints. A linear programming (LP) problem to find an infinitesimal mechanism that maximizes the output displacement is first formulated. The objective function of this LP problem has a penalty term to obtain a sparse solution including small numbers of hinges and members to be removed. It is shown that the dual of this LP problem can be regarded as a plastic limit analysis problem that maximizes the load factor under the equilibrium condition and upper- and lower-bound constraints on the member-end forces of a given frame structure. A heuristic approach is presented to obtain a finite mechanism by solving the LP problem after updating the nodal locations in the direction of inextensional deformation. It is shown in the numerical examples that various planar and spatial mechanisms can be easily found using the proposed method.

Lower-bound-constrained runs in weighted timed automata

We investigate a number of problems related to infinite runs of weighted timed automata (with a single weight variable), subject to lower-bound constraints on the accumulated weight. Closing an open problem from Bouyer et al. (2008), we show that the existence of an infinite lower-bound-constrained run is—for us somewhat unexpectedly—undecidable for weighted timed automata with four or more clocks.This undecidability result assumes a fixed and known initial credit. We show that the related problem of existence of an initial credit for which there exists a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in NEXPTIME. We prove that the universal versions of all those problems (i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.Finally, we extend this study to multi-weighted timed automata: the existence of a feasible run becomes undecidable even for bounded duration, but the existence of initial credits remains decidable (in PSPACE).

Risk index based models for portfolio adjusting problem with returns subject to experts' evaluations

This paper discusses a portfolio adjusting problem with additional risk assets and a riskless asset in the situation where security returns are given by experts' evaluations rather than historical data. Uncertain variables are employed to describe the security returns. Using expected value and risk index as measurements of portfolio return and risk respectively, we propose two portfolio optimization models for an existing portfolio in two cases, taking minimum transaction lot, transaction cost, and lower and upper bound constraints into account. In one case the riskless asset can be both borrowed and lent freely, and in another case the riskless asset can only be lent and the borrowing of riskless asset is not allowed. The adjusting models are converted into their crisp equivalents, enabling the users to solve them with currently available programming solvers. For the sake of illustration, numerical examples in two cases are also provided. The results show that under the same predetermined maximum tolerable risk level the expected return of the optimal portfolio is smaller when the riskless asset can only be lent than when the riskless asset can be both borrowed and lent freely.

On the SCALE Algorithm for Multiuser Multicarrier Power Spectrum Management

This paper studies the successive convex approximation for low complexity (SCALE) algorithm, which was proposed to address the weighted sum rate (WSR) maximized dynamic power spectrum management (DSM) problem for multiuser multicarrier systems. To this end, we first revisit the algorithm, and then present geometric interpretation and properties of the algorithm. A geometric programming (GP) implementation approach is proposed and compared with the low-complexity approach proposed previously. In particular, an analytical method is proposed to set up the default lower-bound constraints added by a GP solver. Finally, numerical experiments are used to illustrate the analysis and compare the two implementation approaches.

Why does leaf nitrogen decline within tree canopies less rapidly than light? An explanation from optimization subject to a lower bound on leaf mass per area

A long-established theoretical result states that, for a given total canopy nitrogen (N) content, canopy photosynthesis is maximized when the within-canopy gradient in leaf N per unit area (N(a)) is equal to the light gradient. However, it is widely observed that N(a) declines less rapidly than light in real plant canopies. Here we show that this general observation can be explained by optimal leaf acclimation to light subject to a lower-bound constraint on the leaf mass per area (m(a)). Using a simple model of the carbon-nitrogen (C-N) balance of trees with a steady-state canopy, we implement this constraint within the framework of the MAXX optimization hypothesis that maximizes net canopy C export. Virtually all canopy traits predicted by MAXX (leaf N gradient, leaf N concentration, leaf photosynthetic capacity, canopy N content, leaf-area index) are in close agreement with the values observed in a mature stand of Norway spruce trees (Picea abies L. Karst.). An alternative upper-bound constraint on leaf photosynthetic capacity (A(sat)) does not reproduce the canopy traits of this stand. MAXX subject to a lower bound on m(a) is also qualitatively consistent with co-variations in leaf N gradient, m(a) and A(sat) observed across a range of temperate and tropical tree species. Our study highlights the key role of constraints in optimization models of plant function.

Three‐dimensional gravity modelling and focusing inversion using rectangular meshes

ABSTRACTRectangular grid cells are commonly used for the geophysical modeling of gravity anomalies, owing to their flexibility in constructing complex models. The straightforward handling of cubic cells in gravity inversion algorithms allows for a flexible imposition of model regularization constraints, which are generally essential in the inversion of static potential field data. The first part of this paper provides a review of commonly used expressions for calculating the gravity of a right polygonal prism, both for gravity and gradiometry, where the formulas of Plouff and Forsberg are adapted. The formulas can be cast into general forms practical for implementation. In the second part, a weighting scheme for resolution enhancement at depth is presented. Modelling the earth using highly digitized meshes, depth weighting schemes are typically applied to the model objective functional, subject to minimizing the data misfit. The scheme proposed here involves a non‐linear conjugate gradient inversion scheme with a weighting function applied to the non‐linear conjugate gradient scheme's gradient vector of the objective functional. The low depth resolution due to the quick decay of the gravity kernel functions is counteracted by suppressing the search directions in the parameter space that would lead to near‐surface concentrations of gravity anomalies. Further, a density parameter transformation function enabling the imposition of lower and upper bounding constraints is employed. Using synthetic data from models of varying complexity and a field data set, it is demonstrated that, given an adequate depth weighting function, the gravity inversion in the transform space can recover geologically meaningful models requiring a minimum of prior information and user interaction.

A sliding time window heuristic for open pit mine block sequencing

The open pit mine block sequencing problem (OPBS) seeks a discretetime production schedule that maximizes the net present value of the orebody extracted from an open-pit mine. This integer program (IP) discretizes the mine’s volume into blocks, imposes precedence constraints between blocks, and limits resource consumption in each time period. We develop a “sliding time window heuristic” to solve this IP approximately. The heuristic recursively defines, solves and partially fixes an approximating model having: (i) fixed variables in early time periods, (ii) an exact submodel defined over a “window” of middle time periods, and (iii) a relaxed submodel in later time periods. The heuristic produces near-optimal solutions (typically within 2% of optimality) for model instances that standard optimization software fails to solve. Furthermore, it produces these solutions quickly, even though our OPBS model enforces standard upper-bounding constraints on resource consumption along with less standard, but important, lower-bounding constraints.

A deterministic annealing algorithm for the minimum concave cost network flow problem

The existing algorithms for the minimum concave cost network flow problems mainly focus on the single-source problems. To handle both the single-source and the multiple-source problem in the same way, especially the problems with dense arcs, a deterministic annealing algorithm is proposed in this paper. The algorithm is derived from an application of the Lagrange and Hopfield-type barrier function. It consists of two major steps: one is to find a feasible descent direction by updating Lagrange multipliers with a globally convergent iterative procedure, which forms the major contribution of this paper, and the other is to generate a point in the feasible descent direction, which always automatically satisfies lower and upper bound constraints on variables provided that the step size is a number between zero and one. The algorithm is applicable to both the single-source and the multiple-source capacitated problem and is especially effective and efficient for the problems with dense arcs. Numerical results on 48 test problems show that the algorithm is effective and efficient.

From Extremal Trajectories to Token Deaths in P-time Event Graphs

In this technical note, we consider the (max,+) model of P-time Event Graphs whose behaviors are defined by lower and upper bound constraints. The extremal trajectories of the system starting from an initial interval are characterized with a particular series of matrices for a given finite horizon. Two dual polynomial algorithms are proposed to check the existence of feasible trajectories. The series of matrices are used in the determination of the maximal horizon of consistency and the calculation of the date of the first token deaths.

A unified transformation function for lower and upper bounding constraints on model parameters in electrical and electromagnetic inversion

Since many geophysical inverse problems are ill-posed, implementation of model parameter constraints is important in reducing solution ambiguity. This paper presents a unified transformation function for enforcing lower and upper bounds on model parameters to restrict model updates during the inversion process such that unrealistic results are suppressed. The transformation is realized using a logarithmic or inverse hyperbolic tangent function on model parameters, and contains the conventional transformations as special cases. It is straightforward to recast the cost functional gradient in terms of the transformed variable. The interval between lower and upper bounds reflects the reliability of a priori information, which may be obtained from well logging and surface geological surveys. Tests on a synthetic crosswell electromagnetic example reveal that the use of bound constraints is effective in deriving valid physical results from inversion.

Treating Uncertain Demand Information in Origin–Destination Matrix Estimation with Traffic Counts

This paper investigates the problem of estimating dynamic origin–destination (O–D) trip matrices in large-scale transportation networks, using automatic link traffic counts and uncertain prior demand information, to facilitate the real-time monitoring of network traffic conditions through intelligent transportation systems. The paper proposes and evaluates the performance of two algorithms to estimate the temporal distribution of trip departures used to produce dynamic O–D matrices, through several test experimentations in a real case study. The accuracy and the computing speed of the proposed algorithms is investigated in relation to different bounds, wherein trip departure estimates are sought to be included, levels of link count availability and network sizes. Results show that an optimization algorithm proposed for the fast estimation of trip departure rates with incorporated lower and upper bound constraints lead to dynamic O–D matrices with considerably increased reliability, in comparison to the al...

Teaching Note—Some Practical Issues with Excel Solver: Lessons for Students and Instructors

Spreadsheets have become the principal software application for teaching decision models in most business schools. In particular, Excel Solver is used extensively for solving and analyzing optimization models. However, there are a number of important issues in using Solver of which many users are unaware. These include the impact of spreadsheet design and cell formatting on Solver reports, handling of lower and upper bound constraints, and dealing with the implications of implicit assumptions in spreadsheet models when interpreting Solver sensitivity reports. Students and instructors, as well as most popular textbooks, rarely pay sufficient attention to these issues. This article summarizes these important issues, provides guidelines for avoiding problems, and offers examples that can be incorporated into the classroom.