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We present an integer program (IP) for urban design that (1) maximizes the floor area; (2) fills building volume with room templates; (3) encodes translational symmetry in urban layout; and (4) constructs economical urban routes. Regardless that integer programming is intensively studied in operational research (OR), its role in solving geometrical and topological problems in urban design was overlooked. Based on a regular grid, our 0–1 IP formulates the sunlight-gain rules, which give urban sites their shapes, especially for residential projects. With predefined plot templates, the gross floor area (volume) within a given site can be maximized under various sunlight requirements. Subsequently, the IP fills each building volume with 2D/3D room templates. Finally, an IP-based algorithm constructs routes that connect all plots and the site’s entrances to public transportation. Both the classical Steiner tree model and the latest coverage network model are extended to create reasonable routes. In addition, this work extends the concept of special ordered sets (SOS) to encode translational symmetry in urban layouts. Encoding layout symmetry can benefit from the solvers’ SOS2 mechanism in the Branch-and-Bound search algorithm. The results indicate that traditional decision making for cities could be partially automated by IP and an abundance of valid solutions are available for designers.

This paper presents an approximated AC model for simultaneous transmission expansion planning (TEP) and reactive power planning by considering wind power investment. Since the full AC-TEP problem is still challenging to solve due to its non-convexity, a linearized model, by means of special ordered set of type 2, is used to represent the mathematical model of the network. The objective function of our problem is considered to be the annualized investment cost of transmission lines, reactive power resources and wind power plants as well as the operation cost of production of electricity and reactive power resources. Wind and load uncertainties are handled by scenario generation. The K-means clustering technique is employed in order to reduce the number of scenarios. The presented work is applied to the 6-bus Garver’s power system, the 24-bus IEEE RTS and the 118-bus IEEE power system. These three different examples illustrate the characteristics of the method. The results and the provided discussions clearly show the effectiveness and robustness as well as the computational efficiency of the proposed stochastic co-planning framework.

District energy systems (DESs) have become a popular form of satisfying comprehensive energy demands for different types of loads in multiple local buildings. For DESs, the operational flexibility could be maintained by energy conversion and storage facilities. This paper proposes a hierarchical optimization framework for leveraging and aggregating the DES flexibility to provide contingency reserves. To characterize and quantify the flexibility in individual DESs, the concept of available reserve profile, which is measured by a set of indices, is established. A two-stage robust optimization (RO) model is developed for calculating the indices, which considers the uncertainties associated with wind power and ambient temperature. The lower stage of the two-stage model is managed by district energy system operators (DESOs) which submit reserve profiles to the district energy system coordinator (DESC) at the upper stage, which is responsible for the coordination process. Correspondingly, information privacy is preserved using a coordinated data-sharing strategy. Using reserve profiles submitted by multiple DESOs, the DESC applies the proposed coordination model to provide a certain reserve capacity schedule to DESs, which satisfies the stated objectives. The coordination model is formulated and solved based on the special ordered set (SOS) technique and particle swarm optimization (PSO) algorithm. A test system is developed to illustrate the technical viability and economic feasibility of the proposed technique.

This work presents a MILP formulation for the optimal operation planning of Multi-Energy Systems serving district heating networks. In particular, the work focuses on Multi-Energy Systems (MES) featuring thermal generation units connected in series and/or parallel, units with limitations on the operating temperature range of inlet/outlet water, and headers in which occurs the non-isothermal mixing of water flows. The model accounts for both dispatchable (e.g. CHP engines) and non-dispatchable (e.g. thermal solar panels) generation units, as well as for the presence of stratified thermal storages. To avoid the nonlinearity and nonconvexity associated to the variable stream temperatures and non-isothermal mixing, each water flow is represented as a linear combination of at most two virtual flows at discrete temperature levels, imposing a Type 2 Special Ordered Set (SOS2) condition to identify the two flows with closest temperatures. The model is used to optimize the operation of an integrated MES designed to serve the electric and thermal loads of a University Campus.

As a natural extension of three-way decisions, this paper presents a novel three-way decision model with order information. First, we do some comparative analysis between ordered three-way decisions and decision-theoretic rough sets, and then present some important properties of the proposed model. Second, a hybrid decision table consisted both of the “order information” and “loss function”, is utilized to solve the ordered three-way decisions with two classification problem. Two order sets (dominating set and dominated set generated by order relation) and three risk strategies (optimistic strategy, equable strategy, pessimistic strategy) are induced to construct the model and design the algorithm of ordered three-way decisions. At last, an illustrative example of salary administration validates the reasonability and effectiveness of our approach.

This study presents a novel networked microgrid (MG)-aided approach for service restoration in power distribution systems. This study considers both dispatchable and non-dispatchable distributed generators (DGs), and energy storage systems. The uncertainty of the customer load demands and DG outputs are modelled in a scenario-based form. A stochastic mixed-integer linear program is formulated with the objective to maximise the served load, while satisfying the operation constraints of the distribution system and MGs. The interaction among MGs is modelled using the type 1 special ordered set. Two approaches are developed and compared: (i) a centralised approach where all MGs are controlled by a distribution system operator, and (ii) a decentralised approach where the distribution system and MGs are managed by different entities. The proposed restoration models are tested on a modified IEEE 123-bus distribution system. The results demonstrate the advantages of leveraging networked MGs to facilitate service restoration.

Rework adversely impacts the performance of building projects. In this study, data were analyzed from 788 construction incidents in 40 Spanish building projects to determine the effects of project and managerial characteristics on rework costs. Finally, regression analysis was used to understand the relationships among contributing factors and to develop a model for rework prediction. Interestingly, the rework prediction model showed that only the original contract value (OCV) and the project location in relation to the company’s headquarters contributed to the regression model. Project type, type of organization, type of contract, and original contract duration (OCD), which represents the magnitude and complexity of a project, were represented by the OCV. This model for rework prediction based on original project conditions enables strategies to be put in place prior to the start of construction, to minimize uncertainties, to reduce impacts on project cost and schedule, and, thus, to improve productivity.

Mixed integer quadratic programs with quadratic constraints (MIQQP) occur frequently in various areas of engineering practice and management science, but most solution methods for this kind of problems are often designed for its special cases. In this paper, we present a simple global optimization algorithm for solving problem (MIQQP). We first convert problem (MIQQP) into an equivalent generalized bilinear programming problem with integer variables (EIQQP). We next show that replacing the quadratic objective and constraint functions with their convex envelopes is dominated by an alternative methodology based on convexifying the range of the bilinear terms on the feasible region. Finally, by incorporating the reduction-correction techniques and sampling strategies into the branch and bound scheme, the proposed algorithm is developed for solving (MIQQP). Convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and MINLPLib2 are carried out to validate the performance of the proposed algorithm.

Branch and bound is a well-known generic method for computing an optimal solution of a single-objective optimization problem. Based on the idea “divide to conquer”, it consists in an implicit enumeration principle viewed as a tree search. Although the branch and bound was first suggested by Land and Doig (1960), the first complete algorithm introduced as a multi-objective branch and bound that we identified was proposed by Kiziltan and Yucaoglu (1983). Rather few multi-objective branch and bound algorithms have been proposed. This situation is not surprising as the contributions on the extensions of the components of branch and bound for multi-objective optimization are recent. For example, the concept of bound sets, which extends the classic notion of bounds, has been mentioned by Villarreal and Karwan (1981). But it was only developed for the first time in 2001 by Ehrgott and Gandibleux, and fully defined in 2007. This paper describes a state-of-the-art of multi-objective branch and bound, which reviews concepts, components and published algorithms. It mainly focuses on the contributions belonging to the class of optimization problems who has received the most of attention in this context from 1983 until 2015: the linear optimization problems with zero-one variables and mixed 0–1/continuous variables. Only papers aiming to compute a complete set of efficient solutions are discussed.

The short-term hydrothermal scheduling (SHS), typically a complicated nonlinear, nonconvex and non-smooth optimization problem, is very important for the economic operation of power systems. Instead of heuristic algorithms popularly used in previous studies, this paper employs a mathematical approach, where a two-stage linear programming with special ordered sets (TLPSOS) is proposed to solve the SHS problem. The nonlinear thermal cost functions and hydropower output functions are approximated by using the special ordered sets. The TLPSOS involves two stages: solve the linearized model in the first stage, and eliminate the linearization errors in the second. Superior to heuristic algorithms, the TLPSOS does not rely on parameters, and can always give stable results. Applied to a widely used hydrothermal system which consists of four hydroplants and three thermal plants, the present method shows its efficiency and strength in obtaining results better than those of previous studies.

The cyclic best-first search (CBFS) strategy is a recent search strategy that has been successfully applied to branch-and-bound algorithms in a number of different settings. CBFS is a modification of best-first search (BFS) that places search tree subproblems into contours which are collections of subproblems grouped in some way, and repeatedly cycles through all non-empty contours, selecting one subproblem to explore from each. In this article, the theoretical properties of CBFS are analyzed for the first time. CBFS is proved to be a generalization of all other search strategies by using a contour definition that explores the same sequence of subproblems as any other search strategy. Further, a bound is proved between the number of subproblems explored by BFS and the number of children generated by CBFS, given a fixed branching strategy and set of pruning rules. Finally, a discussion of heuristic contour-labeling functions is provided, and proof-of-concept computational results for mixed-integer programming problems from the MIPLIB 2010 database are shown. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 64–82, 2017

We provide a polynomial time algorithm that identifies if a given finite ordered set is in the class of d2-collapsible ordered sets. For a d2-collapsible ordered set, the algorithm also determines if the ordered set is connectedly collapsible. Because finite ordered sets of interval dimension 2 are d2-collapsible, in particular, the algorithm determines in polynomial time if a given finite ordered set of interval dimension 2 has the fixed point property. This result is also a first step in investigating the complexity status of the question whether a given collapsible ordered set has the fixed point property.

Frame structures are extensively used in mechanical, civil, and aerospace engineering. Besides generating reasonable designs of frame structures themselves, frame topology optimization may serve as a tool providing us with conceptual designs of diverse engineering structures. Due to its nonconvexity, however, most of existing approaches to frame topology optimization are local optimization methods based on nonlinear programming with continuous design variables or (meta)heuristics allowing some discrete design variables. Presented in this paper is a new global optimization approach to the frame topology optimization with discrete design variables. It is shown that the compliance minimization problem with predetermined candidate cross-sections can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution is then computed with an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to examine computational efficiency of the proposed approach.

The global optimization of integer and mixed integer non-linear problems has a lot of applications in engineering. In this paper a heuristic algorithm is developed using line-up competition and generalized pattern search to solve integer and mixed integer non-linear optimization problems subjected to various linear or nonlinear constraints. Due to its ability to find more than one local or global optimal points, the proposed algorithm is more beneficial for multi-modal problems. The performance of this algorithm is demonstrated through several non-convex integer and mixed integer optimization problems exhibiting good agreement with those reported in the literature. In addition, the convergence time is compared with LCAs' one demonstrating the efficiency and speed of the algorithm. Meanwhile, the constraints are satisfied after passing only a few iterations.

This paper presents a model for a transmission expansion planning (TEP) problem in which both active and reactive power as well as voltage magnitude of buses are considered through lineariz...

To take into account that expert's degrees of certainty are not always comparable, researchers have used partially ordered set of degrees instead of the more traditional linearly (totally) ordered interval (0, 1). In most cases, it is assumed that this partially ordered set is a lattice, i.e., every two elements have the greatest lower bound and the least upper bound. In this paper, we prove a theorem explaining why it is reasonable to require that the set of degrees is a lattice.

Mixed integer control systems are used to model dynamical behavior that can change instantly, for example a driving car with different gears. Changing a gear corresponds to an instant change of the differential equation what is achieved in the model by changing the value of the integer control function. The optimal control of a mixed integer control system by a discretize-then-optimize approach leads to a mixed integer optimization problem that is not differentiable with respect to the integer variables, such that gradient based optimization methods can not be applied. In this work, differentiability with respect to all optimization variables is achieved by reformulating the mixed integer optimal control problem (MIOCP). A fixed integer control function and a time transformation are introduced. The combination of both allows to change the sequence of active differential equations by partially deactivating the fixed integer control function. In contrast to other works, here different fixed integer control functions are taken into account. Advantages of so called control consistent (CC) fixed integer control functions are discussed and confirmed on a numerical example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The set partitioning problem is a well-known NP-hard combinatorial optimization problem, and it is formulated as an integer programming model. This paper proposes an Integer Programming-based Local Search for solving the set partitioning problem. The key point is to solve the set partitioning problem as the set covering problem. First, an initial solution is generated by a simple heuristic for the set covering problem, and then the solution is set as the current solution. Next, the following process is repeated. The original set covering problem is reduced based on the current solution, and the reduced problem is solved by Integer Programming which includes a specific element in the objective function to derive the solution for the set partitioning problem. Experimental results on a set of OR-Library instances show that the proposed algorithm outperforms pure integer programming as well as the existing heuristic algorithms both in solution quality and time.

We consider a balance-constrained stochastic bottleneck spanning tree problem (BCSBSTP) where edge weights are independently distributed but may follow arbitrary continuous distributions. The goal is to minimize a threshold variable that may be exceeded by the maximum edge weight at certain risk, subject to the minimum edge weight being no less than a fixed threshold with a probability guarantee. We characterize these two requirements as chance constraints, which are typically used for bounding the risk of undesirable random outcomes. Given independently distributed edge weights, we reformulate BCSBSTP as a mixed-integer nonlinear program, approximated by two mixed-integer linear programs based on special ordered set of type one (SOS1) and special ordered set of type two (SOS2) variables. By relaxing the probabilistic guarantee on the minimum edge weight in BCSBSTP, we also consider a stochastic bottleneck spanning tree problem (SBSTP), of which optimal tree solutions are approximated via a bisection algorithm in pseudopolynomial time. We demonstrate computational results of our models and algorithms by testing randomly generated instances with edge weights following a diverse set of independent distributions.

This paper studies the mean-risk portfolio optimization problem with nonconvex transaction costs. We employ the conditional value-at-risk (CVaR) as a risk measure. There are a number of studies that aim at efficiently solving large-scale CVaR minimization problems. None of these studies, however, take into account nonconvex transaction costs, which are present in practical situations. To make a piecewise linear approximation of the transaction cost function, we utilized special ordered set type two constraints. Moreover, we devised a subgradient-based cutting plane algorithm to handle a large number of scenarios. This cutting plane algorithm needs to solve a mixed integer linear programming problem in each iteration, and this requires a substantial computation time. Thus, we also devised a two-phase cutting plane algorithm that is even more efficient. Numerical experiments demonstrated that our algorithms can attain near-optimal solutions to large-scale problems in a reasonable amount of time. Especially when rebalancing a current portfolio that is close to an optimal one, our algorithms considerably outperform other solution methods.

Mathematical optimization problems with a goal function, have many applications in various fields like financial sectors, management sciences and economic applications. Therefore, it is very important to have a powerful tool to solve such problems when the main criterion is not linear, particularly fractional, a ratio of two affine functions. In this paper, we propose an exact algorithm for optimizing a linear fractional function over the efficient set of a Multiple Objective Integer Linear Programming (MOILP ) problem without having to enumerate all the efficient solutions. We iteratively add some constraints, that eliminate the undesirable (not interested) points and reduce, progressively, the admissible region. At each iteration, the solution is being evaluated at the reduced gradient cost vector and a new direction that improves the objective function is then defined. The algorithm was coded in MATLAB environment and tested over different instances randomly generated.

The paper deals with the use of special (expanded) branch and bound method, which allows to significantly increase the dimension of the problem, also shows the comparative characteristics of the traditional approach and proposed.

This paper presents an efficient method for finding all solution sets of piecewise-linear interval equations using integer programming. In this method, the problem of finding all solution sets is formulated as a mixed integer programming problem, and it is solved by a high-performance integer programming solver such as CPLEX. It is shown that the proposed method can be implemented easily without writing complicated programs, and that all solution sets are obtained by solving a mixed integer programming problem only once. Numerical examples are given to confirm the effectiveness of the proposed method.

A direct-search derivative-free Matlab optimizer for bound-constrained problems is described, whose remarkable features are its ability to handle a mix of continuous and discrete variables, a versatile interface as well as a novel self-training option. Its performance compares favorably with that of NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search), a well-known derivative-free optimization package. It is also applicable to multilevel equilibrium- or constrained-type problems. Its easy-to-use interface provides a number of user-oriented features, such as checkpointing and restart, variable scaling, and early termination tools.

We consider a class of mixed integer programs where the problem is convex except for a vector of discrete variables. Two methods based on the Alternating Direction Method of Multipliers (ADMM) are presented. The first, which has appeared in the recent literature, duplicates the discrete variable, with one copy allowed to vary continuously. This results in a simple projection, or rounding, to determine the discrete variable at each iteration. We introduce an alternate method, whereby part of the objective is replaced by a new variable instead. When the objective satisfies a certain condition, this allows the update of the discrete variables to be handled separately for each one, thus maintaining linear complexity of this update, while incorporating some of the objective into the update. Initial comparisons on examples for which both methods are applicable show that the latter exhibits clear improvements in both performance and run-time.

In this paper, a new branch-and-cut algorithm for mixed integer bi-level programming is proposed. For achieving this purpose, a historical perspective of the development of enumeration methods in the field of bi-level linear programming is considered. Then, we present some obstacles for using branch and bound method based on them, and an algorithm is developed to solve for mixed integer bi-level problem. Finally, we use a preference function to determine the choice of branching and specialized cuts in a branch and cut tree. Computational results are reported and compared favorably to those of previous methods and then implications discussed. The results show that not only the proposed algorithm can find high quality solutions for solving a number of the problems, but also it is competitive with other famous published algorithms.

The set covering problem (SCP) is one of representative combinatorial optimization problems, which is defined as the problem of covering the m-rows by a subset of the n-columns at minimal cost. This paper proposes a method utilizing Integer Programming-based Local Search (IPbLS) to solve the set covering problem. IPbLS is a kind of local search technique in which the current solution is improved by searching neighborhood solutions. Integer programming is used to generate neighborhood solution in IPbLS. The effectiveness of the proposed algorithm has been tested on OR-Library test instances. The experimental results showed that IPbLS could search for the best known solutions in all the test instances. Especially, I confirmed that IPbLS could search for better solutions than the best known solutions in four test instances. ∙제1저자 : 황준하 ∙교신저자 : 황준하 ∙투고일 : 2014. 7. 22, 심사일 : 2014. 8. 11, 게재확정일 : 2014. 9. 11. * 금오공과대학교 컴퓨터공학과(Dept. of Computer Engineering, Kumoh National Institute of Technology) ※ 본 연구는 금오공과대학교 학술연구비에 의하여 연구된 논문임 14 Journal of The Korea Society of Computer and Information October 2014 ▸

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.

Multiobjective combinatorial optimization deals with problems considering more than one viewpoint or scenario. The problem of aggregating multiple criteria to obtain a globalizing objective function is of special interest when the number of Pareto solutions becomes considerably large or when a single, meaningful solution is required. Ordered weighted average or ordered median operators are very useful when preferential information is available and objectives are comparable since they assign importance weights not to specific objectives but to their sorted values. In this paper, ordered weighted average optimization problems are studied from a modeling point of view. Alternative integer programming formulations for such problems are presented and their respective domains studied and compared. In addition, their associated polyhedra are studied and some families of facets and new families of valid inequalities presented. The proposed formulations are particularized for two well-known combinatorial optimization problems, namely, shortest path and minimum cost perfect matching, and the results of computational experiments presented and analyzed. These results indicate that the new formulations reinforced with appropriate constraints can be effective for efficiently solving medium to large size instances.

We report and analyze the results of our computational testing of branch-and-cut for the complementarity-constrained optimization problem (CCOP). Besides the MIP cuts commonly present in commercial optimization software, we used inequalities that explore complementarity constraints. To do so, we generalized two families of cuts proposed earlier by de Farias, Johnson, and Nemhauser that had never been tested computationally. Our test problems consisted of linear, binary, and general integer programs with complementarity constraints. Our results on the use of complementarity cuts within a major commercial optimization solver show that they are of critical importance to tackling difficult CCOP instances, typically reducing the computational time required to solve them tremendously.

Most real-world optimization problems are multiobjective by nature, involving noncomparable objectives. Many of these problems can be formulated in terms of a set of linear objective functions that should be simultaneously optimized over a class of linear constraints. Often there is the complicating factor that some of the variables are required to be integral. The resulting class of problems is named multiobjective mixed integer programming (MOMIP) problems. Solving these kinds of optimization problems exactly requires a method that can generate the whole set of nondominated points (the Pareto-optimal front). In this paper, we first give a survey of the newly developed branch and bound methods for solving MOMIP problems. After that, we propose a new branch and bound method for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables. The proposed method is able to find the full set of nondominated points. It is tested on a large number of problem instances, from six different classes of MOMIP problems. The results reveal that the developed biobjective branch and bound method performs better on five of the six test problems, compared with a generic two-phase method. At this time, the two-phase method is the most preferred exact method for solving MOMIP problems with two criteria and binary variables. This paper was accepted by Dimitris Bertsimas, optimization.

We are interested in methods to solve mixed-integer nonlinear optimal control problems constrained by ordinary differential equations and combinatorial constraints on some of the control functions. To solve these problems we use a first discretize, then optimize approach to get a specially structured mixed-integer nonlinear program (MINLP). We decompose this MINLP into a nonlinear program (NLP) and a mixed-integer linear program (MILP), which is called the combinatorial integral approximation problem (CIAP). Previous results guarantee an integer gap for the MINLP depending on the objective function value of the CIAP. The focus of this study is the analysis of the CIAP and of a tailored branch-and-bound method. We link the huge computational gains compared to state-of-the-art MILP solvers to an analysis of subproblems on the branching tree. To this end we study properties of the Lagrangian relaxation of the CIAP. Special focus is given to special ordered set constraints that are present due to an outer con...

Geometric branch-and-bound techniques are well-known solution algorithms for non-convex continuous global optimization problems with box constraints. Several approaches can be found in the literature differing mainly in the bounds used. The aim of this paper is to extend geometric branch-and-bound methods to mixed integer optimization problems, i.e. to objective functions with some continuous and some integer variables. Mixed-integer non-linear and non-convex optimization problems are extremely hard, containing several classes of NP-hard problems as special cases. We identify for which type of mixed integer non-linear problems our method can be applied efficiently, derive several bounding operations and analyze their rates of convergence theoretically. Moreover, we show that the accuracy of any algorithm for solving the problem with fixed integer variables can be transferred to the mixed integer case. Our results are demonstrated theoretically and experimentally using the truncated Weber problem and the p -median problem. For both problems we succeed in finding exact optimal solutions.

An order on a set of centered intervals is introduced and is proved to be linear. An optimization problem is formulated over a set of centered intervals. The branch and bound method is proposed and substantiated to solve this problem. A number of theorems that substantiate estimates in the branch and bound method are proved.

We consider a mixed integer set that results from the intersection of a simple mixed integer set with a vertex packing set from a conflict graph. This set arises as a relaxation of the feasible set of mixed integer problems such as inventory routing problems. We derive families of strong valid inequalities that consider the structures of the simple mixed integer set and the vertex packing set simultaneously.

The classical harmony search algorithm (HSA) can only be used to solve the unconstrained optimization problems with continuous decision variables. Therefore, the classical HSA is not suitable for solving an engineering optimization problem with mixed discrete variables. In order to improve the classical HSA, an engineering method for dealing with mixed discrete decision variables is introduced and an exact non-differentiable penalty function is used to transform the constrained optimization design model into an unconstrained mathematical model. Based on above improvements, a program of improved HSA is designed and it can be used for solving the constrained optimization design problems with continuous variables, integer variables and non-equidistant discrete variables. Finally, an optimization design example of single-stage cylindrical-gear reducer with mixed-discrete variables is given. The example shows that the designed program runs steadily and the proposed method is effective in engineering design.

We propose an equivalent reduction of the quantile optimization problem with a discrete distribution of random parameters to a partially integer programming problem of large dimension. The number of integer (Boolean) variables in this problem equals the number of possible values for the random parameters vector. The resulting problems can be solved with standard discrete optimization software. We consider applications to quantile optimization of a financial portfolio and show results of numerical experiments.

SUMMARY This paper presents a comprehensive formulation for the security constrained optimal power flow (SCOPF) problem considering valve loading effect, multiple fuel option, and prohibited operating zones of units as well as alternating current network modeling and contingency constraints. Also, the SCOPF formulation includes the integer variables, such as discrete transformer tap settings, in addition to continuous variables, such as generation of units. Thus, the suggested SCOPF model is a mixed integer, nonlinear, non-convex, and non-smooth optimization problem. To solve this problem, a new solution method composed of Benders decomposition and special ordered sets is presented. The proposed formulation decomposes the problem into a master problem and a sub-problem. The master problem relaxes the nonlinear constraints of the model using a convex linear outer approximation based on the concept of special ordered sets, whereas the sub-problem contains the nonlinear and non-convex SCOPF formulation with fixed integer and binary variables. To show the effectiveness of the proposed solution method, it is tested on the well-known test systems and compared with several other recently published solution methods. These comparisons confirm the validity of the developed approach. Copyright © 2013 John Wiley & Sons, Ltd.

This work addresses the correction and improvement of Mavrotas and Diakoulaki's branch and bound algorithm for mixed 0-1 multiple objective linear programs. We first elaborate the issues encountered by the original algorithm and then propose a corrected version for the biobjective case using an exact representation of the nondominated set associated with an appropriate update procedure. Then we introduce several improvements using better bound sets and branching strategies and finally present some experiments to study the effectiveness of our propositions.

Engineering design problems often involve non-linear criterion functions, including inequality and equality constraints, and a mixture of discrete and continuous design variables. Optimization approaches entail substantial challenges when solving such an all-inclusive design problem. In this paper, a modification of the Particle Swarm Optimization (PSO) algorithm is presented, which can adequately address system constraints while dealing with mixed-discrete variables. Continuous search (particle motion), as in conventional PSO, is implemented as the primary search strategy; subsequently, the discrete variables are updated using a deterministic nearest-feasible-vertex criterion. This approach is expected to alleviate the undesirable difference in the rates of evolution of discrete and continuous variables. The premature stagnation of candidate solutions (particles) due to loss of diversity is known to be one of the primary drawbacks of the basic PSO dynamics. To address this issue in high dimensional design problems, a new adaptive diversity-preservation technique is developed. This technique characterizes the population diversity at each iteration. The estimated diversity measure is then used to apply (i) a dynamic repulsion away from the best global solution in the case of continuous variables, and (ii) a stochastic update of the discrete variables. For performance validation, the Mixed-Discrete PSO algorithm is applied to a wide variety of standard test problems: (i) a set of 9 unconstrained problems, and (ii) a comprehensive set of 98 Mixed-Integer Nonlinear Programming (MINLP) problems. We also explore the applicability of this algorithm to a large scale engineering design problem---wind farm layout optimization.

In this paper a mixed integer set resulting from the intersection of a single constrained mixed 0-1 set with the vertex packing set is investigated. This set arises as a subproblem of more general mixed integer problems such as inventory routing and facility location problems. Families of strong valid inequalities that take into account the structure of the simple mixed integer set and that of the vertex packing set simultaneously are introduced. In particular, the well-known mixed integer rounding inequality is generalized to the case where incompatibilities between binary variables are present. Exact and heuristic algorithms are designed to solve the separation problems associated to the proposed valid inequalities. Preliminary computational experiments show that these inequalities can be useful to reduce the integrality gaps and to solve integer programming problems.

This paper presents a novel networked microgrids (MGs)-aided approach for service restoration in a power distribution network. A mixed-integer linear program (MILP) is formulated with two objectives, i.e., maximizing the loads picked up and minimizing the restoration time, while satisfying the operation constraints of the distribution system and MGs. The interaction among MGs is modelled as a MILP problem by using the type 1 special ordered set (SOS1) to maintain the power supply to critical loads in MGs and pick up on-outage loads outside MGs. The main contribution of this paper is to leverage networked MGs to facilitate service restoration. The model is tested on a modified IEEE 123-node distribution test system to validate the proposed scheme. The results show the effectiveness of applying networked MGs to service restoration.

Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.