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Showing papers on "Nonlinear programming published in 2007"


Journal ArticleDOI
TL;DR: The main focus will be on the different approaches to perform robust optimization in practice including the methods of mathematical programming, deterministic nonlinear optimization, and direct search methods such as stochastic approximation and evolutionary computation.

1,435 citations


Journal ArticleDOI
TL;DR: This paper presents fields of application, focus on solution approaches, and makes the connection with MPECs (Mathematical Programs with Equilibrium Constraints), a branch of mathematical programming of both practical and theoretical interest.
Abstract: This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

1,364 citations


Journal ArticleDOI
TL;DR: Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Abstract: (2007). Nonlinear Programming Theory and Algorithms. Technometrics: Vol. 49, No. 1, pp. 105-105.

1,317 citations


Journal ArticleDOI
TL;DR: This work presents a systematic method of distributed algorithms for power control that is geometric-programming-based and shows that in the high Signal-to- interference Ratios (SIR) regime, these nonlinear and apparently difficult, nonconvex optimization problems can be transformed into convex optimized problems in the form of geometric programming.
Abstract: In wireless cellular or ad hoc networks where Quality of Service (QoS) is interference-limited, a variety of power control problems can be formulated as nonlinear optimization with a system-wide objective, e.g., maximizing the total system throughput or the worst user throughput, subject to QoS constraints from individual users, e.g., on data rate, delay, and outage probability. We show that in the high Signal-to- interference Ratios (SIR) regime, these nonlinear and apparently difficult, nonconvex optimization problems can be transformed into convex optimization problems in the form of geometric programming; hence they can be very efficiently solved for global optimality even with a large number of users. In the medium to low SIR regime, some of these constrained nonlinear optimization of power control cannot be turned into tractable convex formulations, but a heuristic can be used to compute in most cases the optimal solution by solving a series of geometric programs through the approach of successive convex approximation. While efficient and robust algorithms have been extensively studied for centralized solutions of geometric programs, distributed algorithms have not been explored before. We present a systematic method of distributed algorithms for power control that is geometric-programming-based. These techniques for power control, together with their implications to admission control and pricing in wireless networks, are illustrated through several numerical examples.

906 citations


Journal ArticleDOI
TL;DR: The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables.
Abstract: The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables. It uses OptQuest, a commercial implementation of scatter search developed by OptTek Systems, Inc., to provide starting points for any gradient-based local solver for nonlinear programming (NLP) problems. This solver seeks a local solution from a subset of these points, holding discrete variables fixed. The procedure is motivated by our desire to combine the superior accuracy and feasibility-seeking behavior of gradient-based local NLP solvers with the global optimization abilities of OptQuest. Computational results include 155 smooth NLP and mixed integer nonlinear program (MINLP) problems due to Floudas et al. (1999), most with both linear and nonlinear constraints, coded in the GAMS modeling language. Some are quite large for global optimization, with over 100 variables and 100 constraints. Global solutions to almost all problems are found in a small number of local solver calls, often one or two.

631 citations


Journal ArticleDOI
TL;DR: This work presents a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing as a finite-dimensional convex optimization problem as a second-order cone programming problem.
Abstract: We present a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing. Our main contribution is the formulation of the trajectory optimization problem, which has nonconvex control constraints, as a finite-dimensional convex optimization problem, specifically as a second-order cone programming problem. Second-order cone programming is a subclass of convex programming, and there are efficient second-order cone programming solvers with deterministic convergence properties. Consequently, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications.

482 citations


Journal ArticleDOI
TL;DR: An algorithm is presented for wheeled mobile robot trajectory generation that achieves a high degree of generality and efficiency and is efficient enough to use in real time due to its use of nonlinear programming techniques that involve searching the space of parameterized vehicle controls.
Abstract: An algorithm is presented for wheeled mobile robot trajectory generation that achieves a high degree of generality and efficiency. The generality derives from numerical linearization and inversion of forward models of propulsion, suspension, and motion for any type of vehicle. Efficiency is achieved by using fast numerical optimization techniques and effective initial guesses for the vehicle controls parameters. This approach can accommodate such effects as rough terrain, vehicle dynamics, models of wheel-terrain interaction, and other effects of interest. It can accommodate boundary and internal constraints while optimizing an objective function that might, for example, involve such criteria as obstacle avoidance, cost, risk, time, or energy consumption in any combination. The algorithm is efficient enough to use in real time due to its use of nonlinear programming techniques that involve searching the space of parameterized vehicle controls. Applications of the presented methods are demonstrated for planetary rovers.

375 citations


Journal ArticleDOI
TL;DR: The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the spectral projected gradient method for solving the subproblems.
Abstract: Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the constant positive linear dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the spectral projected gradient method for solving the subproblems. Problems of this type with more than $3 \times 10^6$ variables and $ 14 \times 10^6$ constraints are solved in this way, using moderate computer time. All the codes are available at http://www.ime.usp.br/$\sim$egbirgin/tango/.

373 citations


Journal ArticleDOI
TL;DR: A relaxation method is described which yields an easily computable upper bound on the optimal solution of portfolio selection, and a heuristic method for finding a suboptimal portfolio which is based on solving a small number of convex optimization problems.
Abstract: We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.

344 citations


Book
08 Jan 2007
TL;DR: In this article, the authors propose a simplex method for robust optimization in finance, using linear programming, nonlinear programming, and Quadratic programming, with the use of robust optimization tools.
Abstract: 1. Introduction 2. Linear programming: theory and algorithms 3. LP models: asset/liability cash flow matching 4. LP models: asset pricing and arbitrage 5. Nonlinear programming: theory and algorithms 6. NLP volatility estimation 7. Quadratic programming: theory and algorithms 8. QP models: portfolio optimization 9. Conic optimization tools 10. Conic optimization models in finance 11. Integer programming: theory and algorithms 12. IP models: constructing an index fund 13. Dynamic programming methods 14. DP models: option pricing 15. DP models: structuring asset backed securities 16. Stochastic programming: theory and algorithms 17. SP models: value-at-risk 18. SP models: asset/liability management 19. Robust optimization: theory and tools 20. Robust optimization models in finance Appendix A. Convexity Appendix B. Cones Appendix C. A probability primer Appendix D. The revised simplex method Bibliography Index.

261 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider a supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers (DCs), where customers face random demand, and each DC maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves.

Journal ArticleDOI
TL;DR: In this article, a new evolutionary algorithm known as bacteria foraging is applied for solving the multiobjective multivariable problem, with the UPFC location, its series injected voltage, and the transformer tap positions as the variables.
Abstract: Optimal location and control of a unified power flow controller (UPFC) along with transformer taps are tuned with a view to simultaneously optimize the real power losses and voltage stability limit (VSL) of a mesh power network. This issue is formulated as a nonlinear equality and inequality constrained optimization problem with an objective function incorporating both the real power loss and VSL. A new evolutionary algorithm known as bacteria foraging is applied for solving the multiobjective multivariable problem, with the UPFC location, its series injected voltage, and the transformer tap positions as the variables. For a single objective of only real power loss, the same problem is also solved with interior point successive linearization program (IPSLP) technique using the LINPROG command of MATLAB. A comparison between the two suggests the superiority of the proposed algorithm. A cost effectiveness analysis of UPFC installation vis-agrave-vis loss reduction is carried out to establish the benefit of investment in a UPFC

Journal ArticleDOI
TL;DR: This paper proposes and compares different one-dimensional maps as chaotic search patterns in the constraint nonlinear optimization problems and applies them in specific optimization algorithm (weighted gradient direction based chaos optimization algorithm) and compares them based on numerical simulation results.

Journal ArticleDOI
27 Aug 2007
TL;DR: In this paper, a constructive heuristic algorithm aimed at obtaining an excellent quality solution for this problem is presented, where an interior point method is employed to solve nonlinear programming problems during the solution steps of the algorithm.
Abstract: An optimisation technique to solve transmission network expansion planning problem, using the AC model, is presented. This is a very complex mixed integer nonlinear programming problem. A constructive heuristic algorithm aimed at obtaining an excellent quality solution for this problem is presented. An interior point method is employed to solve nonlinear programming problems during the solution steps of the algorithm. Results of the tests, carried out with three electrical energy systems, show the capabilities of the method and also the viability of using the AC model to solve the problem.

Journal ArticleDOI
TL;DR: Various objectives of reactive power planning are reviewed and various optimization models, identified as optimal power flow model, security-constrained OPF model, and SCOPF with voltage-stability consideration are discussed.
Abstract: The key of reactive power planning (RPP), or Var planning, is the optimal allocation of reactive power sources considering location and size. Traditionally, the locations for placing new Var sources were either simply estimated or directly assumed. Recent research works have presented some rigorous optimization-based methods in RPP. This paper will first review various objectives of RPP. The objectives may consider many cost functions such as variable Var cost, fixed Var cost, real power losses, and fuel cost. Also considered may be the deviation of a given voltage schedule, voltage stability margin, or even a combination of different objectives as a multi-objective model. Secondly, different constraints in RPP are discussed. These different constraints are the key of various optimization models, identified as optimal power flow (OPF) model, security-constrained OPF (SCOPF) model, and SCOPF with voltage-stability consideration. Thirdly, the optimization-based models will be categorized as conventional algorithms, intelligent searches, and fuzzy set applications. The conventional algorithms include linear programming, nonlinear programming, mixed-integer nonlinear programming, etc. The intelligent searches include simulated annealing, evolutionary algorithms, and tabu search. The fuzzy set applications in RPP address the uncertainties in objectives and constraints. Finally, this paper will conclude the discussion with a summary matrix for different objectives, models, and algorithms.

Journal ArticleDOI
TL;DR: Integer programming and network flow-based lower-bounding methods that can solve moderately large instances of the WTA problem optimally and obtain almost optimal solutions of fairly large instances within a few seconds are suggested.
Abstract: The weapon-target assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research. This problem consists of optimally assigning n weapons to m targets so that the total expected survival value of the targets after all the engagements is minimal. The WTA problem can be formulated as a nonlinear integer programming problem and is known to be NP-complete. No exact methods exist for the WTA problem that can solve even small-size problems (for example, with 20 weapons and 20 targets). Although several heuristic methods have been proposed to solve the WTA problem, due to the absence of exact methods, no estimates are available on the quality of solutions produced by such heuristics. In this paper, we suggest integer programming and network flow-based lower-bounding methods that we obtain using a branch-and-bound algorithm for the WTA problem. We also propose a network flow-based construction heuristic and a very large-scale neighborhood (VLSN) search algorithm. We present computational results of our algorithms, which indicate that we can solve moderately large instances (up to 80 weapons and 80 targets) of the WTA problem optimally and obtain almost optimal solutions of fairly large instances (up to 200 weapons and 200 targets) within a few seconds.

Journal ArticleDOI
TL;DR: This work presents combinatorial methods to preprocess these matrices to establish more favorable numerical properties for the subsequent factorization in a sparse direct LDLT factorization method where the pivoting is restricted to static supernode data structures.
Abstract: Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints.

Journal ArticleDOI
TL;DR: An optimization method using the process synthesis approach to design an RO system has been developed in this article, which offers extensive flexibility towards optimizing various types of RO system and thus may be used for the selection of the optimal structural and operating schemes.

Journal ArticleDOI
TL;DR: A novel nonlinear neural network (NN) predictive control strategy based on the new tent-map chaotic particle swarm optimization (TCPSO) is presented to enhance the convergence and accuracy of the TCPSO.
Abstract: In this letter, a novel nonlinear neural network (NN) predictive control strategy based on the new tent-map chaotic particle swarm optimization (TCPSO) is presented. The TCPSO incorporating tent-map chaos, which can avoid trapping to local minima and improve the searching performance of standard particle swarm optimization (PSO), is applied to perform the nonlinear optimization to enhance the convergence and accuracy. Numerical simulations of two benchmark functions are used to test the performance of TCPSO. Furthermore, simulation on a nonlinear plant is given to illustrate the effectiveness of the proposed control scheme

Proceedings ArticleDOI
01 May 2007
TL;DR: Simulation results show that solutions obtained by this algorithm are very close to lower bounds obtained via relaxation, thus suggesting that the solution produced by the algorithm is near-optimal.
Abstract: Software defined radio (SDR) capitalizes advances in signal processing and radio technology and is capable of reconfiguring RF and switching to desired frequency bands. It is a frequency-agile data communication device that is vastly more powerful than recently proposed multi-channel multi-radio (MC-MR) technology. In this paper, we investigate the important problem of multi-hop networking with SDR nodes. For such network, each node has a pool of frequency bands (not necessarily of equal size) that can be used for communication. The uneven size of bands in the radio spectrum prompts the need of further division into sub-bands for optimal spectrum sharing. We characterize behaviors and constraints for such multi-hop SDR network from multiple layers, including modeling of spectrum sharing and sub-band division, scheduling and interference constraints, and flow routing. We give a formal mathematical formulation with the objective of minimizing the required network-wide radio spectrum resource for a set of user sessions. Since such problem formulation falls into mixed integer non-linear programming (MINLP), which is NP-hard in general, we develop a lower bound for the objective by relaxing the integer variables and linearization. Subsequently, we develop a near-optimal algorithm to this MINLP problem. This algorithm is based on a novel sequential fixing procedure, where the integer variables are determined iteratively via a sequence of linear programming. Simulation results show that solutions obtained by this algorithm are very close to lower bounds obtained via relaxation, thus suggesting that the solution produced by the algorithm is near-optimal.

01 May 2007
TL;DR: The LTMads-PB is a useful practical extension of the earlier L TMads-EB algorithm, particularly in the common case for real problems where no feasible point is known, and it does as well when feasible points are known.
Abstract: We propose a new constraint-handling approach for general constraints that is applicable to a widely used class of constrained derivative-free optimization methods. As in many methods that allow infeasible iterates, constraint violations are aggregated into a single constraint violation function. As in filter methods, a threshold, or barrier, is imposed on the constraint violation function, and any trial point whose constraint violation function value exceeds this threshold is discarded from consideration. In the new algorithm, unlike the filter method, the amount of constraint violation subject to the barrier is progressively decreased adaptively as the iteration evolves. We test this progressive barrier (PB) approach versus the extreme barrier (EB) with the generalized pattern search (Gps) and the lower triangular mesh adaptive direct search (LTMads) methods for nonlinear derivative-free optimization. Tests are also conducted using the Gps-filter, which uses a version of the Fletcher-Leyffer filter approach. We know that Gps cannot be shown to yield kkt points with this strategy or the filter, but we use the Clarke nonsmooth calculus to prove Clarke stationarity of the sequences of feasible and infeasible trial points for LTMads-PB. Numerical experiments are conducted on three academic test problems with up to 50 variables and on a chemical engineering problem. The new LTMads-PB method generally outperforms our LTMads-EB in the case where no feasible initial points are known, and it does as well when feasible points are known. which leads us to recommend LTMads-PB. Thus the LTMads-PB is a useful practical extension of our earlier LTMads-EB algorithm, particularly in the common case for real problems where no feasible point is known. The same conclusions hold for Gps-PB versus Gps-EB.

Journal ArticleDOI
TL;DR: A probabilistic bi-level linear multi-objective programming problem and its application in enterprise-wide supply chain planning problem where (1) market demand, (2) production capacity of each plant and (3) resource available to all plants for each product are random variables and the constraints may consist of joint probability distributions or not.

Journal ArticleDOI
TL;DR: An online calibration approach that jointly estimates demand and supply parameters of dynamic traffic assignment (DTA) systems is presented and empirically validated through an extensive application.
Abstract: An online calibration approach that jointly estimates demand and supply parameters of dynamic traffic assignment (DTA) systems is presented and empirically validated through an extensive application The problem can be formulated as a nonlinear state-space model Because of its nonlinear nature, the resulting model cannot be solved by the Kalman filter, and therefore, nonlinear extensions need to be considered The following three extensions to the Kalman filtering algorithm are presented: 1) the extended Kalman filter (EKF); 2) the limiting EKF (LimEKF); and 3) the unscented Kalman filter The solution algorithms are applied to the on-line calibration of the state-of-the-art DynaMIT DTA model, and their use is demonstrated in a freeway network in Southampton, UK The LimEKF shows accuracy that is comparable to that of the best algorithm but with vastly superior computational performance The robustness of the approach to varying weather conditions is demonstrated, and practical aspects are discussed

Journal ArticleDOI
TL;DR: The development goes beyond a simple exercise of applying scatter search to this class of problems, but presents innovative mechanisms to obtain a good balance between intensification and diversification in a short-term search horizon.
Abstract: Scatter search is a population-based method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in 1960s for combining decision rules and problem constraints such as the surrogate constraint method, scatter search uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we develop a general purpose heuristic for a class of nonlinear optimization problems. The procedure is based on the scatter search methodology and treats the objective function evaluation as a black box, making the search algorithm context-independent. Most optimization problems in the chemical and bio-chemical industries are highly nonlinear in either the objective function or the constraints. Moreover, they usually present differential-algebraic systems of constraints. In this type of problem, the evaluation of a solution or even the feasibility test of a set of values for the decision variables is a time-consuming operation. In this context, the solution method is limited to a reduced number of solution examinations. We have implemented a scatter search procedure in Matlab (Mathworks, 2004) for this special class of difficult optimization problems. Our development goes beyond a simple exercise of applying scatter search to this class of problems, but presents innovative mechanisms to obtain a good balance between intensification and diversification in a short-term search horizon. Computational comparisons with other recent methods over a set of benchmark problems favor the proposed procedure.

Journal ArticleDOI
TL;DR: TIMP is an R package for modeling multiway spectroscopic measurements that fits separable nonlinear models using partitioned variable projection, a variant of the variable projection algorithm that is described here for the first time.
Abstract: TIMP is an R package for modeling multiway spectroscopic measurements. The package allows for the simultaneous analysis of datasets collected under different experimental conditions in terms of a wide variety of parametric models. Models arising in spectroscopy data analysis often have some parameters that are intrinstically nonlinear, and some parameters that are conditionally linear on estimates of the nonlinear parameters. TIMP fits such separable nonlinear models using partitioned variable projection, a variant of the variable projection algorithm that is described here for the first time. The of the partitioned variable projection algorithm allows fitting many models for spectroscopy datasets using much less memory as compared to under the standard variable projection algorithm that is implemented in nonlinear optimization routines (e.g., the plinear option of the R function nls), as is shown here. An overview of modeling with TIMP is also given that includes several case studies in the application of the package.

Journal ArticleDOI
TL;DR: In this paper, a general low-thrust trade analysis tool is developed based on a global search for local indirect method solutions, and an efficient propagator is implemented with an implicit "bang-bang" thrusting structure with an a priori unknown number of switching times.
Abstract: The low-thrust spacecraft trajectory problem can be reduced to only a few parameters using calculus of variations and the well-known primer vector theory. This low dimensionality combined with the extraordinary speed of modern computers allows for rapid exploration of the parameter space and invites opportunities for global optimization. Accordingly, a general low-thrust trade analysis tool is developed based on a global search for local indirect method solutions. An efficient propagator is implemented with an implicit "bang-bang" thrusting structure that accommodates an a priori unknown number of switching times. An extension to the standard adjoint control transformation is introduced that provides additional physical insight and control over the anticipated evolution of the thrust profile. The uniformly random search enjoys a perfect linear speedup for parallel implementation. The method is applied specifically on multi revolution transfers in the Jupiter-Europa and Earth-moon restricted three body problems. In both cases, thousands of solutions are found in a single parallel run. The result is a global front of Pareto optimal solutions across the competing objectives of flight time and final mass.


Journal ArticleDOI
TL;DR: This paper addresses the optimal placement of static Var compensators in a transmission network in such a manner that its loading margin is maximized by using a Benders decomposition technique within a restart framework.
Abstract: This paper addresses the optimal placement of static Var compensators (SVCs) in a transmission network in such a manner that its loading margin is maximized. A multi scenario framework that includes contingencies is considered. This problem is formulated as a nonlinear programming problem that includes binary decisions, i.e., variables to decide the actual placement of the SVCs. Given the mixed-integer nonconvex nature of this problem, a Benders decomposition technique within a restart framework is used. Detailed numerical simulations on realistic electric energy systems demonstrate the appropriate behavior of the proposed technique. Conclusions are duly drawn.

Journal ArticleDOI
TL;DR: It was indicated that the proposed linearization method was effective in dealing with IFNP problems; uncertainties can be communicated into optimization process and generate reliable solutions for decision variables and objectives; the decision alternatives can be obtained by adjusting different combinations of the decision variables within their solution intervals.

Journal ArticleDOI
TL;DR: In this paper, a GA-Benders' decomposition (GA-BD) method is proposed for solving the power generation expansion planning problem with emission control, and an application of the proposed GA-BD method is discussed and concluded.
Abstract: The power generation expansion planning (PGEP) problem is a large-scale mixed integer nonlinear programming (MINLP) problem cited as one of the most complex optimization problems. In this paper, an application of a new efficient methodology for solving the power generation expansion planning problem is presented. A comprehensive planning production simulation model is introduced toward formulating into an MINLP model. The model evaluates the most economical investment planning for additional thermal power generating units of the optimal mix for long-term power generation expansion planning with emission controls, regarding to the incorporated environmental costs, subject to the integrated requirements of power demands, power capacities, loss of load probability (LOLP) levels, locations, and environmental limitations for emission controls. A GA-heuristic-based method called GA-Benders' decomposition (GA-BD) is proposed for solving this complex problem. Finally, an application of the proposed GA-BD method is discussed and concluded.