Combined Relaxation Method Research Articles

An Integer Mathematical Programming Model for Production Scheduling Problem in Open-Pit Mines under Grade Uncertainty and Solving Using the Firefly Algorithm

Long-term production scheduling in open-pit mines is a crucial issue in mining planning and determines the distribution of cash flow throughout the life of the mine. The purpose of the planning is to maximize the net present value by taking into account all operational constraints such as slope, mixing of different grades, mineral production, and extraction capacity. The uncertainties associated with model data play an important role in optimizing long-term production plans. Among the uncertainties, grade uncertainty plays a major role. In this paper, hybrid models are presented by the Lagrangian relaxation (LR) method, augmented Lagrangian relaxation (ALR) method, and firefly algorithm (FA) to solve the long-term production scheduling problem of open-pit mines with the assumption of deterministic and also considering the grade uncertainty. The firefly algorithm is used to update the Lagrange multipliers. The newly proposed approaches are based on optimizing Lagrangian multipliers and comparing them with the results of combined Lagrangian relaxation method and augmented Lagrangian relaxation with the Genetic Algorithm (GA), and the traditional sub-gradient (SG) method. For solving and validating the obtained model, Chadarmelo iron ore mine is considered as a suitable case study. The results of the case study show that the combined strategy (ALR-FA) can provide a near-optimal solution over other methods such that, over a given period, the net present value using the proposed hybrid approach is 20.11% higher than the traditional method is available. Also, the CPU speed of the proposed model is 4.7% more than the other methods.

Thermodynamic insights into n-alkanes phase change materials for thermal energy storage

n-Alkanes have been widely used as phase change materials (PCMs) for thermal energy storage applications because of their exceptional phase transition performance, high chemical stability, long term cyclic stability and non-toxicity. However, the thermodynamic properties, especially heat capacity, of n-alkanes have rarely been comprehensively investigated in a wide temperature range, which would be insufficient for design and utilization of n-alkanes-based thermal energy storage techniques. In this study, the thermal properties of n-alkanes (C18H38-C22H46), such as thermal stability, thermal conductivity, phase transition temperature and enthalpy were systematically studied by different thermal analysis and calorimetry methods, and compared with previous results. Thermodynamic property of these n-alkanes was studied in a wide temperature range from 1.9 K to 370 K using a combined relaxation (Physical Property Measurement System, PPMS), differential scanning and adiabatic calorimetry method, and the corresponding thermodynamic functions, such as entropy and enthalpy, were calculated based on the heat capacity curve fitting. Most importantly, the heat capacities and related thermodynamic functions of n-heneicosane and n-docosane were reported for the first time in this work, as far as we know. This research work would provide accurate and reliable thermodynamic properties for further study of n-alkanes-based PCMs for thermal energy storage applications.

An improved projection method for solving generalized variational inequality problems

ABSTRACTIn this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is obtained by projecting the current iterate point onto a half-space. This half-space can separate strictly the current iterate point from the solution set of GVIP. Moreover, this method works without needing the current point belongs to the feasible set. Comparing with methods in Konnov [A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998;80(2):239–252] and Fang and Chen [Subgradient extragradient algorithm for solving multi-valued variational inequality. Appl Math Comput. 2014;229(3-4):123–130], our method can get rid of an auxiliary procedure in each iteration which is used to ensure the current iterate point belongs to feasible set. Consequently, our method is more simpler than those algorithms. The global convergence is proved under mild assumptions. Numerical results show that this method is much more efficient than the method in Li and He [An algorithm for generalized variational inequality with pseudomonotone mapping. J Comput Appl Math. 2009;228:212–218].

Preferential Solvation of an Asymmetric Redox Molecule

The fundamental correlations between solubility and solvation structure for the electrolyte system comprising N-(ferrocenylmethyl)-N,N-dimethyl-N-ethylammonium bistrifluoromethylsulfonimide (Fc1N112-TFSI) dissolved in a ternary carbonate solvent mixture is analyzed using combined NMR relaxation and computational methods. Probing the evolution of the solvent–solvent, ion–solvent and ion–ion interactions with an increase in solute concentration provides a molecular level understanding of the solubility limit of the Fc1N112-TFSI system. An increase in solute concentration leads to pronounced Fc1N112-TFSI contact-ion pair formation by diminishing solvent–solvent and ion–solvent type interactions. At the solubility limit, the precipitation of solute is initiated through agglomeration of contact-ion pairs due to overlapping solvation shells.

A Strongly Convergent Combined Relaxation Method in Hilbert Spaces

We consider a combined relaxation method for variational inequalities in a Hilbert space setting. Methods of this class are known to solve finite-dimensional variational inequalities under mild monotonicity type assumptions, whereas in Hilbert space strong monotonicity is the standard assumption for strong convergence. Here, we relax this condition and show strong convergence of such a method, when strong monotonicity holds only on a subspace of finite co-dimension. Thus, the method applies to semi-coercive unilateral boundary value problems in mathematical physics.

Study of structural and dynamic characteristics of copper(II) amino acid complexes in solutions by combined EPR and NMR relaxation methods.

Structural features and dynamical behaviour of the copper(ii) bis-complexes with glycine, d-alanine, d-valine, l-serine, l-aspartic acid, l-glutamic acid, l-lysine, l-proline, and sarcosine were studied by combined EPR and NMR relaxation methods. The cis and trans isomers were unambiguously assigned and characterized by EPR data. It was found that addition of a salt background has an influence on the cis-trans isomer equilibrium in favour of the formation of the cis isomer. By comparison of NMRD, DFT computations, and structural data it was shown that only one water molecule is coordinated in the axial position of these complexes. The increased exchange rates of this molecule found for Cu(l-Asp)2(2-), Cu(l-Glu)2(2-), Cu(l-LysH)2(2+), and Cu(l-Pro)2 were attributed to its pushing out by side chain groups of the ligands. By simulation of NMRD profiles an increase of lifetimes of the copper(ii) 2nd coordination sphere water molecules was revealed in the presence of additional carboxylic, alcoholic, or ammonium groups of the ligands, as well as the pyrrolidine ring of proline. The very short lifetimes of the 2nd coordination sphere water molecules (4-13 ps at 298 K) were explained in terms of the Frank-Wen structural model by the existence of cavities which draw in quickly enough water molecules from the 2nd coordination sphere.

A modified combined relaxation method for non-linear convex variational inequalities

We consider a class of non-linear problems which is intermediate between equilibrium and variational inequality ones and has many applications. Unlike the usual variational inequality it involves two non-linear mappings, which need not be differentiable. We propose a class of iterative methods for this problem, which converge to a solution under weakened monotonicity type assumptions. This method is simpler essentially in comparison with those for the corresponding non-linear equilibrium problems.

LAGRANGIAN RELAXATION-BASED PARTICLE SWARM OPTIMIZATION FOR UNIT COMMITMENT PROBLEM

This paper presents an application of combined Lagrangian relaxation (LR) method and particle swarm optimization (PSO) technique for the Unit Commitment (UC) problem. This hybrid technique uses LR as the main algorithm. The LR method provides a fast solution, but it may suffer from numerical convergence and solution quality problem. The proposed algorithm uses PSO to update the Lagrangian multipliers and improve the performance of LR method. In this paper, ramp rate limit is also considered besides taking minimum up time and minimum down time of thermal units. The proposed method is implemented for solving two examples of UC test problems. The study results have shown that the approach developed is feasible and efficient.

Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces

In this paper, we study a class of general monotone equilibrium problems in a real Hilbert space which involves a monotone differentiable bifunction. For such a bifunction, a skew-symmetric type property with respect to the partial gradients is established. We suggest to solve this class of equilibrium problems with the modified combined relaxation method involving an auxiliary procedure. We prove the existence and uniqueness of the solution to the auxiliary variational inequality in the auxiliary procedure. Further, we prove also the weak convergence of the modified combined relaxation method by virtue of the monotonicity and the skew-symmetric type property.

An exact algorithm for cost minimization in series reliability systems with multiple component choices

In this paper, we present an exact method for cost minimization problems in series reliability systems with multiple component choices. The problem can be modelled as a nonlinear integer programming problem with a nonseparable constraint function. The method is of a combined Lagrangian relaxation and linearization method. A Lagrangian bound is obtained by solving the dual of a separable subproblem. An alternative lower bound is derived by 0–1 linearization method. A special cut-and-partition scheme is proposed to reduce the duality gap, thus ensuring the convergence of the method. Computational results are reported to show the efficiency of the method.

Combined Relaxation Method for Mixed Equilibrium Problems

We consider a general class of equilibrium problems which involve a single-valued mapping and a nonsmooth bifunction. Such mixed equilibrium problems are solved with a combined relaxation method using an auxiliary iteration of a splitting-type method for constructing a separating hyperplane. We prove the convergence of the method under the assumption that the dual of the mixed equilibrium problem is solvable. Convergence rates are also derived.

Splitting-type method for systems of variational inequalities

We consider a system of variational inequalities with multivalued mappings, which can be viewed as an extension of constrained primal–dual variational inequalities. We propose to solve this problem with the help of a class of combined relaxation and splitting methods. We establish a convergence result for these methods for the case where the dual system is solvable. We also give several examples of applications to saddle point problems in optimization, Nash equilibrium problems in game theory, and economic equilibrium problems.

Solution of degenerate parabolic variational inequalities with convection

Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated - saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

A Class of Combined Relaxation Methods for Decomposable Variational Inequalities

Combined relaxation methods are convergent to a solution of variational inequality problems under rather mild assumptions and admit various auxiliary procedures within their two-level structure. In this work, we consider ways to construct decomposition schemes within one class of combined relaxation methods, which maintain useful convergence properties. An application to primal-dual variational inequality problems is also given.

A Combined Relaxation Method for Nonlinear Variational Inequalities

When applied to variational inequalities, combined relaxation (CR) methods are convergent under mild assumptions. Namely, the underlying mapping need not be strictly monotone. In this paper, we describe a class of CR methods for nonlinear variational inequality problems (NVI), which involve two, rather than one, nonlinear mappings and a nonsmooth convex function. We establish a convergence result for the CR method in the monotone case and also show that it attains a linear rate of convergence under the additional strong monotonicity assumption. Implementation issues are also discussed.

A Combined Relaxation Method for a Class of Nonlinear Variational Inequalities

In this paper, we describe a class of combined relaxation methods for the non strictly monotone nonlinear variational inequality problem, which involves a max-type convex function. This method is readily implementable and attains a linear rate of convergence under certain additional assumptions.

Combined Relaxation Method for Monotone Equilibrium Problems

A scalar equilibrium problem which involves a monotone differentiable cost bifunction is considered. For such bifunction, a skew-symmetric type property with respect to the partial gradients is established. This property enables us to construct a new combined relaxation method and essentially simplify its line search procedure. An application to an inverse equilibrium problem is also presented.

Combined Relaxation Methods for Variational

Combined Relaxation Methods for Variational

A combined relaxation method for decomposable variational inequalities *

An iterative method based on combining, modifying and generalizing different relaxation subgradient methods is proposed for solving variational inequality problems. For certain structured problems this method leads to a decomposition scheme. Convergence of the method is proved under weak assumptions. In particular, the main mapping need not be single-valued or monotone.

A combined relaxation method for variational inequalities with nonlinear constraints

A simple iterative method for solving variational inequalities with a set-valued, nonmonotone mapping and a convex feasible set is proposed. This set can be defined by nonlinear functions. The method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization. Also a modification of the averaging method for the problem under consideration is proposed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.