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This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Global optimality condition based on canonical duality theory is derived. Integrating the dual solutions with the KKT conditions, we are able to obtain the approximate solutions or global solutions easily.

To assess the degree of practical implementation of the optimization principle when comparing options for plans compiled in different ways, this paper proposes a comprehensive indicator of the effectiveness of planned calculations. The advantage of the indicator is that its value is proportional to the magnitude of potential losses from incomplete and incomplete use of available resources, that is, those factors that symbolize the loss of resources in the economic planning process, but have not yet served as criteria for the quality of planning decisions. Therefore, the fact that the optimization method makes it possible to improve (reduce) the value of these indicators with the same volumes of available production resources allows us to conclude that the structural optimization method is very effective and promising in solving production problems of linear programming and in the process of economic planning.

We address the problem of multiple local optima commonly arising in optimization problems for multi-agent systems, where objective functions are nonlinear and nonconvex. For the class of coverage control problems, we propose a systematic approach for escaping a local optimum, rather than randomly perturbing controllable variables away from it. We show that the objective function for these problems can be decomposed to facilitate the evaluation of the local partial derivative of each node in the system and to provide insights into its structure. This structure is exploited by defining "boosting functions" applied to the aforementioned local partial derivative at an equilibrium point where its value is zero so as to transform it in a way that induces nodes to explore poorly covered areas of the mission space until a new equilibrium point is reached. The proposed boosting process ensures that, at its conclusion, the objective function is no worse than its pre-boosting value. However, the global optima cannot be guaranteed. We define three families of boosting functions with different properties and provide simulation results illustrating how this approach improves the solutions obtained for this class of distributed optimization problems.

This article is concerned with the question of constructing effcient multigrid preconditioners for the linear systems arising when applying semismooth Newton methods to large-scale linear-quadratic optimization problems constrained by smoothing operators with box-constraints on the controls. It is shown that, for certain discretizations of the optimization problem, the linear systems to be solved at each semismooth Newton iteration reduce to inverting principal minors of the Hessian of the associated unconstrained problem. As in the case when box-constraints on the controls are absent, the multigrid preconditioner introduced here is shown to increase in quality as the mesh-size decreases, resulting in a number of iterations that decreases with mesh-size. However, unlike the unconstrained case, the spectral distance between the preconditioners and the Hessian is shown to be of suboptimal order in general.

This article is concerned with the question of constructing effcient multigrid preconditioners for the linear systems arising when applying semismooth Newton methods to large-scale linear-quadratic optimization problems constrained by smoothing operators with box-constraints on the controls. It is shown that, for certain discretizations of the optimization problem, the linear systems to be solved at each semismooth Newton iteration reduce to inverting principal minors of the Hessian of the associated unconstrained problem. As in the case when box-constraints on the controls are absent, the multigrid preconditioner introduced here is shown to increase in quality as the mesh-size decreases, resulting in a number of iterations that decreases with mesh-size. However, unlike the unconstrained case, the spectral distance between the preconditioners and the Hessian is shown to be of suboptimal order in general.

We study the relationship between unit-distance representations and Lovasz theta number of graphs, originally established by Lovasz. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization of the hypersphere number of a graph and the related optimization problems. The generalized problem involves finding the smallest ellipsoid of a given shape which contains a unit-distance representation of the graph. We prove that arbitrary positive semidefinite forms describing the ellipsoids yield NP-hard problems.

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.

We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in $O(n\log n)$ worst-case time, where $n$ is the number of disks. In the minimum-separation problem, we are given $n$ unit disks and two points $s$ and $t$, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting $s$ to $t$ intersects some of the retained disks. We present a new algorithm solving this problem in $O(n^2\log^3 n)$ worst-case time and its implementation.

Let $\xx_n=(x_1,\ldots,x_n)$ and $f\in \R[\xx_n,k]$. The problem of finding all $k_0$ such that $f(\xx_n,k_0)\ge 0$ on $\mathbb{R}^n$ is considered in this paper, which obviously takes as a special case the problem of computing the global infimum or proving the semi-definiteness of a polynomial. For solving the problems, we propose a simplified Brown's CAD projection operator, \Nproj, of which the projection scale is always no larger than that of Brown's. For many problems, the scale is much smaller than that of Brown's. As a result, the lifting phase is also simplified. Some new algorithms based on \Nproj\ for solving those problems are designed and proved to be correct. Comparison to some existing tools on some examples is reported to illustrate the effectiveness of our new algorithms.

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann's direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.

This paper proposes different methods for finding the global minimum of concave function subject to quadratic separable constraints. THe first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

An algorithm is presented for solution of dynamic optimization problems which are nonlinear in the state variables and linear in the control variables. It is shown that the optimal control is bang-bang. A nominal bang-bang solution is found which satisfies the system equations and constraints, and influence functions are generated which check the optimality of the solution. Nonlinear optimization (gradient search) techniques are used to find the optimal solution. The algorithm is used to find a minimum time acceleration for a turbofan engine.

It has been recently argued that adiabatic quantum optimization would fail in solving NP-complete problems because of the occurrence of exponentially small gaps due to crossing of local minima of the final Hamiltonian with its global minimum near the end of the adiabatic evolution. Using perturbation expansion, we analytically show that for the NP-hard problem of maximum independent set there always exist adiabatic paths along which no such crossings occur. Therefore, in order to prove that adiabatic quantum optimization fails for any NP-complete problem, one must prove that it is impossible to find any such path in polynomial time.

The purpose of this article is to implement different methods of meta-heuristic algorithms to solve five engineering problems. these engineering problems have been optimized using five meta-heuristic algorithms of firefly, colonial competition, frog, ants and gray wolf with the aim of reducing the costs of engineering problems. in each of the algorithms, a dynamic adaptive factor is introduced to balance the convergence rate and absolute optimal search ability by adjusting the search speed during the search process. Investigations show that in each of the algorithms techniques are used to leave the local optimum, which makes the answers converge to the absolute optimum. To evaluate the quality and accuracy of the algorithms, the sensitivity test and the comparison of the convergence numbers for the results of the implementation of each algorithm on the data have been used. The results show that the firefly algorithm in spring tension problem, frog algorithm in three-bar truss problem, the colonial competition algorithm in the speed reducer and gear design problems, and the gray wolf algorithm in the pressure tank problem provided more accurate performance in finding the absolute optimum. . In fact, these algorithms make it easy to achieve the optimal solution by generating a random population, creating a neighborhood and choosing the best neighbor, provided that the constraints of the variables of the problem are satisfied. As a result, this paper shows that any meta-heuristic algorithm can perform better in a specific engineering problem, depending on the type of problem and environmental conditions.

This paper presents an innovative approach in finding an optimal solution of multi-modal and multivariable function for global optimization problems that involve complex and inefficient second derivatives. Artificial bees colony (ABC) algorithm possessed good exploration search, but the major weakness at its exploitation stage. The proposed algorithms improved the weakness of ABC algorithm by hybridizing with the most effective gradient based methods which are Davidon-Flecher-Powell (DFP) and Broyden-Flecher-Goldfarb-Shanno (BFGS) algorithms. Its distinguished features include maximizing the employment of possible information related to the objective function obtained at previous iterations. The proposed algorithms have been tested on a large set of benchmark global optimization problems and it has shown a satisfactory computational behaviour and it has succeeded in enhancing the algorithm to obtain the solution for global optimization problems. 

The Barzilai–Borwein method offers efficient step sizes for large-scale un-constrained optimization problems. However, it may not guarantee global convergence for nonquadratic objective functions. Simulated annealing-based on Barzilai–Borwein (SABB) method addresses this issue by in-corporating a simulated annealing rule. This work proposes a novel step-size strategy for the SABB method, referred to as the SABBm method. Furthermore, we introduce two stabilized variants: SABBstab and SABBmstab. SABBstab combines a simulated annealing rule with a sta-bilization step to ensure convergence. SABBmstab builds upon SABBstab, incorporating the modified step size derived from the SABBm method. The effectiveness and competitiveness of the proposed methods are demon-strated through numerical experiments on CUTEr benchmark problems.

This invention is a novel high-speed neural network based processor for solving the 'traveling salesman' and other global optimization problems. It comprises a novel hybrid architecture employing a binary synaptic array whose embodiment incorporates the fixed rules of the problem, such as the number of cities to be visited. The array is prompted by analog voltages representing variables such as distances. The processor incorporates two interconnected feedback networks, each of which solves part of the problem independently and simultaneously, yet which exchange information dynamically.

Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. During solution of the multidisciplinary problems several issues were encountered. This paper lists four issues and discusses the strategies adapted for their resolution: (1) The optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. (2) Optimum solutions obtained were infeasible for aircraft and air-breathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. (3) Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. (4) The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through six problems: (1) design of an engine component, (2) synthesis of a subsonic aircraft, (3) operation optimization of a supersonic engine, (4) design of a wave-rotor-topping device, (5) profile optimization of a cantilever beam, and (6) design of a cvlindrical shell. The combined effort of designers and researchers can bring the optimization method from academia to industry.

Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. During solution of the multidisciplinary problems several issues were encountered. This paper lists four issues and discusses the strategies adapted for their resolution: (1) The optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. (2) Optimum solutions obtained were infeasible for aircraft and air-breathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. (3) Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. (4) The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through six problems: (1) design of an engine component, (2) synthesis of a subsonic aircraft, (3) operation optimization of a supersonic engine, (4) design of a wave-rotor-topping device, (5) profile optimization of a cantilever beam, and (6) design of a cvlindrical shell. The combined effort of designers and researchers can bring the optimization method from academia to industry.

Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. During solution of the multidisciplinary problems several issues were encountered. This paper lists four issues and discusses the strategies adapted for their resolution: (1) The optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. (2) Optimum solutions obtained were infeasible for aircraft and air-breathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. (3) Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. (4) The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through six problems: (1) design of an engine component, (2) synthesis of a subsonic aircraft, (3) operation optimization of a supersonic engine, (4) design of a wave-rotor-topping device, (5) profile optimization of a cantilever beam, and (6) design of a cvlindrical shell. The combined effort of designers and researchers can bring the optimization method from academia to industry.

Introduction. Reliability-Based Design Optimization (RBDO) model reduces the structural weight in uncritical regions, does not only provide an improved design but also a higher level of confidence in the design. Materials and Methods. The classical RBDO approach can be carried out in two separate spaces: the physical space and the normalized space. Since very many repeated researches are needed in the above two spaces, the computational time for such an optimization is a big problem. An efficient method called Optimum Safety Factor (OSF) method is developed and successfully put to use in several engineering applications. Research Results. A numerical application on a large scale problem under  fatigue  loading  shows  the  efficiency of the developed RBDO method relative to the Deterministic Design Optimization (DDO). The efficiency of the OSF method is also extended to multiple failure modes to control several out-put parameters, such as structural volume and damage criterion. Discussion and Conclusions. The simplified implementation framework of the OSF strategy consists of a single optimization problem to evaluate the design point, and a direct evaluation of the optimum solution considering OSF formulations. It provides designers with efficient solutions that should be economic, satisfying a required reliability level with a reduced computing time.

Recent years have witnessed an increasing interest in Wireless Sensor Networks (WSNs) for various applications such as environmental monitoring and military field surveillance. WSN have a number of sensor nodes that communicate wirelessly and it deployed to gather data for various environments. But it has issue with the energy efficiency of sensor nodes and network lifetime along with packet scheduling. The target coverage problem is another problem hence the overall network performance is reduced significantly. In this research, new Markov Chain Monte Carlo (MCMC) is introduced which solves the energy efficiency of sensor nodes in HWSN. At initially graph model is modeled to represent distributed and heterogeneous (HWSNs) with each vertex representing the assignment of a sensor nodes in a subset. Modified Bat Optimization (MBAT) is proposed to maximize the number of Disjoint Connected Covers (DCC) and K Coverage (KC) known as MBAT-MDCCKC. Based on echolocation capability from the MBAT, the bat seeks an optimal path on the construction routing for packet transmission that maximizes the MDCCKC. MBAT bats thus focus on finding one more connected covers and avoids creating subsets particularly. It designed to increase the search efficiency and hence energy efficiency is improved prominently. The proposed MBAT-MDCCKC approach has been applied to a variety of HWSNs. The results show that the MBAT-MDCCKC approach is efficient and successful in finding optimal results for maximizing the lifetime of HWSNs. Experimental results show that, proposed MBAT-MDCCKC approach performs better than, TFMGA, Bacteria Foraging Optimization (BFO) based approach, Ant Colony Optimization (ACO) method, and the performance of the MBAT-MDCCKC approach is closer to the energy conserving strategy. P. V. Ravindranath | Dr. D. Maheswari"Optimal Energy and Network Lifetime Maximization using a Modified Bat Optimization Algorithm (MBAT) under Coverage Constrained Problems over Heterogeneous Wireless Sensor Networks" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-6 , October 2017, URL: http://www.ijtsrd.com/papers/ijtsrd4731.pdf Article URL: http://www.ijtsrd.com/computer-science/computer-network/4731/optimal-energy-and-network-lifetime-maximization-using-a-modified-bat-optimization-algorithm-mbat-under-coverage-constrained-problems-over-heterogeneous-wireless-sensor-networks/p-v-ravindranath

We will survey recent work in the design of approximation algorithms for several discrete stochastic optimization problems, with a particular focus on 2-stage problems with recourse. In each of the problems we discuss, we are given a probability distribution over inputs, and the aim is to find a feasible solution that minimizes the expected cost of the solution found (with respect to the input distribution); an approximation algorithm finds a solution that is guaranteed to be nearly optimal. Among the specific problems that we shall discuss are stochastic generalizations of the traditional deterministic facility location problem, a simple single-machine scheduling problem, and the traveling salesman problem.These results build on techniques initially developed in the context of deterministic approximation, including rounding approaches, primal-dual algorithms, as well as a simple random sampling technique. Furthermore, although the focus of this stream of work was for discrete optimization problems, new insights for solving 2-stage stochastic linear programming problems were gained along the way. ©2008 Microsoft Corporation. All rights reserved.

Recent developments in mathematical optimization are substantially enhancing the scope and power of logistics planning systems. Based on these advances, successful applications of sophisticated mathematical optimization logistics systems are occurring worldwide. This paper briefly discusses some of these applications and advances. (Author)

We approach a class of discrete event simulation-based optimization problems using optimality in probability, an approach which yields what is termed a "champion solution". Compared to the traditional optimality in expectation, this approach favors the solution whose actual performance is more likely better than that of any other solution; this is an effective alternative to the traditional optimality sense, especially when facing a dynamic and nonstationary environment. Moreover, using optimality in probability is computationally promising for a class of discrete event simulation-based optimization problems, since it can reduce computational complexity by orders of magnitude compared to general simulation-based optimization methods using optimality in expectation. Accordingly, we have developed an "Omega Median Algorithm" in order to effectively obtain the champion solution and to fully utilize the efficiency of well-developed off-line algorithms to further facilitate timely decision making. An inventory control problem with nonstationary demand is included to illustrate and interpret the use of the Omega Median Algorithm, whose performance is tested using simulations.

Decomposition of large problems into a hierarchic pyramid of subproblems was proposed in the literature as a means for optimization of engineering systems too large for all-in-one optimization. This decomposition was established heuristically. The dynamic programming (DP) method due to Bellman was augmented with an optimum sensitivity analysis that provides a mathematical basis for the above decomposition, and overcomes the curse of dimensionality that limited the original formulation of DP. Numerical examples are cited.

Decomposition of large problems into a hierarchic pyramid of subproblems was proposed in the literature as a means for optimization of engineering systems too large for all-in-one optimization. This decomposition was established heuristically. The dynamic programming (DP) method due to Bellman was augmented with an optimum sensitivity analysis that provides a mathematical basis for the above decomposition, and overcomes the curse of dimensionality that limited the original formulation of DP. Numerical examples are cited.

Decomposition of large problems into a hierarchic pyramid of subproblems was proposed in the literature as a means for optimization of engineering systems too large for all-in-one optimization. This decomposition was established heuristically. The dynamic programming (DP) method due to Bellman was augmented with an optimum sensitivity analysis that provides a mathematical basis for the above decomposition, and overcomes the curse of dimensionality that limited the original formulation of DP. Numerical examples are cited.

Recent developments in mathematical optimization are substantially enhancing the scope and power of logistics planning systems. Based on these advances, successful applications of sophisticated mathematical optimization logistics systems are occurring worldwide. This paper briefly discusses some of these applications and advances. (Author)

Recent developments in mathematical optimization are substantially enhancing the scope and power of logistics planning systems. Based on these advances, successful applications of sophisticated mathematical optimization logistics systems are occurring worldwide. This paper briefly discusses some of these applications and advances. (Author)

Recent developments in mathematical optimization are substantially enhancing the scope and power of logistics planning systems. Based on these advances, successful applications of sophisticated mathematical optimization logistics systems are occurring worldwide. This paper briefly discusses some of these applications and advances. (Author)

The paper formulates models of defense problems which are convex programs having the mathematical properties treated in previous papers. The models include several strategic forces planning models and two general purpose forces planning models.

The conjugate gradient (CG) method is an optimization technique known for its rapid convergence; it has blossomed into significant developments and applications. Numerous variations of CG methods have emerged to en-hance computational efficiency and address real-world challenges. In this work, a novel conjugate gradient method is introduced to solve nonlinear unconstrained optimization problems. Based on the combination of PRP (Polak–Ribière–Polyak), HRM (Hamoda–Rivaie–Mamat) and NMFR (new modified Fletcher–Reeves) algorithms, our method produces a descent di-rection without depending on any line search. Moreover, it enjoys global convergence under mild assumptions and is applied successfully on various standard test problems as well as image processing. The numerical results indicate that the proposed method outperforms several existing methods in terms of efficiency.

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

This project studied and developed simultaneous generalized hill climbing (SGHC) algorithms as an algorithmic framework for information sharing in discrete optimization problems. This framework has been used to gain new insights into neighborhood structure designs that allow different neighborhood functions to share information when using the same heuristic applied to the same problem. The results reported from this project introduce the SGHC algorithm framework for information sharing across sets of related discrete optimization problems, provide guidelines on how to use and to design neighborhood functions that results in effective performance of local search algorithms, and describe how tabu search can be effectively used to improve the performance of generalized hill climbing algorithms. Extensive computationally results are reported on a large variety of test bed, large-scale, real-world discrete optimization problems. The primary application for this research were a military combat search and rescue problems, where several possible search and rescue strategies must be considered to determine the optimal strategy, and a homeland security aviation security baggage screening problems, where several different baggage screening strategies at a set of airports must be considered to determine the optimal strategy for the entire system. Both these problems are intractable due, in part, to the exponentially large number of possible solutions that exist and must be evaluated to identify those that are optimal.

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover, a type of integer programming (IP) problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of $\alpha$-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical-mechanical model with a one-parameter family. Statistical-mechanical analyses reveal for $\alpha=2$ that the LP optimal solution is typically equal to that given by the IP below the critical average degree $c=e$ in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result $c=1$, and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with $\alpha\geq 3$, minimum vertex covers on $\alpha$-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree $c=e/(\alpha-1)$ where the replica symmetry is broken.

In this paper we present a novel method for solving optimization problems governed by partial differential equations. Existing methods are gradient information in marching toward the minimum, where the constrained PDE is solved once (sometimes only approximately) per each optimization step. Such methods can be viewed as a marching techniques on the intersection of the state and costate hypersurfaces while improving the residuals of the design equations per each iteration. In contrast, the method presented here march on the design hypersurface and at each iteration improve the residuals of the state and costate equations. The new method is usually much less expensive per iteration step since, in most problems of practical interest, the design equation involves much less unknowns that that of either the state or costate equations. Convergence is shown using energy estimates for the evolution equations governing the iterative process. Numerical tests show that the new method allows the solution of the optimization problem in a cost of solving the analysis problems just a few times, independent of the number of design parameters. The method can be applied using single grid iterations as well as with multigrid solvers.

This project has two objectives. The first is to determine whether linear programming techniques can improve performance when handling design optimization problems with a large number of design variables and constraints relative to the feasible directions algorithm. The second purpose is to determine whether using the Kreisselmeier-Steinhauser (KS) function to replace the constraints with one constraint will reduce the cost of total optimization. Comparisons are made using solutions obtained with linear and non-linear methods. The results indicate that there is no cost saving using the linear method or in using the KS function to replace constraints.

Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrainted optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.

Trajectory physics optimization for boundary value problem solution

Dynamic resource allocation (DRA) problems constitute an important class of dynamic stochastic optimization problems that arise in a variety of important real-world applications. DRA problems are notoriously difficult to solve to optimality since they frequently combine stochastic elements with intractably large state and action spaces. Although the artificial intelligence and operations research communities have independently proposed two successful frameworks for solving dynamic stochastic optimization problems Monte Carlo tree search (MCTS) and rolling horizon optimization (RHO), respectively the relative merits of these two approaches are not well understood. In this paper, we adapt both MCTS and RHO to two problems a problem inspired by tactical wildlife management and a classical problem involving the control of queueing networks and undertake an extensive computational study comparing the two methods on large scale instances of both problems in terms of both the state and the action spaces. We show that both methods are able to greatly improve on a baseline, problem-specific heuristic. On smaller instances, the MCTS and RHO approaches perform comparably, but the RHO approach outperforms MCTS as the size of the problem increases for a fixed computational budget.

Work accomplished over the first three years (1972-75) may be summarized as follows: (1) An extensive study of models for optimally scheduling lots of N products on a single processor under various demand process structures. (2) Solution of a problem of scheduling N jobs on parallel processors to minimize a penalty function based on job due dates when there are no precedence relations among jobs. (3) Two studies treating the efficient solution of minimum cost flow problems. (4) Research on methodologies for discrete optimization related to (a) Tree search in methods of implicit enumeration, (b) Circumventing the 'curse of dimensionality' in dynamic programming, and (c) The reduction method of integer programming for a specialized version of the generalized assignment problem. (5) Surveys of past and current research in (a) Scheduling multiple processors, (b) Flow networks, (c) Project planning networks, (d) Basic concepts used in branch-and-bound, and (e) Production planning.

The Fletcher-Powell version of the Davidon variable metric unconstrained minimization technique is described. Equations that have been used successfully with the Davidon-Fletcher-Powell penalty function technique for solving constrained minimization problems and the advantages and disadvantages of using them are discussed. The experience gained in the behavior of the method while iterating is also related.

The data association problem in multi-target tracking has been formulated and solved as a multidimensional assignment problem. Extensive simulations have been performed to demonstrate speed and robustness of these algorithms.

Two classes of mathematical programs with optimization problems in the constraints have recently been studied by two of the authors. The first class involves mathematical programs in the constraints, and the second class involves max-min problems in the constraints. A computational technique has been developed and shown to be effective in solving problems of the first class. The authors show that the computational technique can be applied to problems of the apparently wider second class.

A genetic algorithm approach suitable for solving multi-objective optimization problems is described and evaluated using a series of simple model problems. Several new features including a binning selection algorithm and a gene-space transformation procedure are included. The genetic algorithm is suitable for finding pareto optimal solutions in search spaces that are defined by any number of genes and that contain any number of local extrema. Results indicate that the genetic algorithm optimization approach is flexible in application and extremely reliable, providing optimal results for all optimization problems attempted. The binning algorithm generally provides pareto front quality enhancements and moderate convergence efficiency improvements for most of the model problems. The gene-space transformation procedure provides a large convergence efficiency enhancement for problems with non-convoluted pareto fronts and a degradation in efficiency for problems with convoluted pareto fronts. The most difficult problems --multi-mode search spaces with a large number of genes and convoluted pareto fronts-- require a large number of function evaluations for GA convergence, but always converge.

Based on a method introduced by Leitmann [Internat. J. Non-Linear Mech. {\bf 2} (1967), 55--59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.

Evolution Strategies are inspired in biology and part of a larger research field known as Evolutionary Algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.

A common problem addressed in Managerial and Cost Accounting classes is that of selecting an optimal production mix given scarce resources. That is, if a firm produces a number of different products, and is faced with scarce resources (e.g., limitations on labor, materials, or machine time), what combination of products yields the greatest profit to the firm? Solver, an optimization package included within Microsoft Excel (or Optimizer in Quattro Pro), is an ideal vehicle by which to analyze these problems. In most cost or managerial accounting texts, students are asked to address this type of question when there is only one scarce resource (e.g., "Material X"); such problems can be readily solved "by hand". In the case of two or more scarce resources, students are usually referred to their management science classes and Linear Programming packages such as LINDO for further enlightenment, with the comment that such matters are beyond the scope of an accounting text. The purpose of this paper is to illustrate how the Solver package in Microsoft Excel can be easily used to solve optimization problems in management accounting. Although not as powerful or flexible as stand-alone packages such as LINDO, Solver's presence within a universally available spreadsheet package makes it an extraordinarily powerful teaching tool. Instead of parameters being entered into the optimization problem as constants, they can be expressed as functions of other spreadsheet cells. This interactive structure allows an instructor (or student) to create complex production environments where it can be illustrated how minor changes in one aspect of the production environment can flow through and have a profound impact on optimal production schedules.