A method for determining optimal multipliers in the surrogate constraint method

Surrogate constraint analysis has been applied effectively to a variety of combinatorial optimization problems, as a foundation for both exact and heuristic methods. In the heuristic domain, surrogate constraint methods are particularly suited to the creation of associated learning procedures and to the application of probabilistic decisions. We show that these approaches are natural and effective for satisfiability (SAT) problems. Added motivation comes from observing that the current best exact and heuristic procedures for multidimensional knapsack problems are provided independently by surrogate constraint methods and probabilistic methods that use memory and learning structures (derived from tabu search). We show that the SAT problem can be formulated as a special instance of a binary-choice multidimensional knapsack problem (or equivalently, a binary-choice generalized covering problem), and demonstrate how surrogate constraint analysis can be specialized in a particularly convenient way to exploit the structure of this problem. Our approach incorporates simple (first order) instances of adaptive memory structures characteristic of tabu search implementations, to give a learning effect to guide the search. This use of memory adds a dimension to the solution process that has not adequately been examined in the past. We find that the combination of surrogate constraint analysis and simple learning proves more effective than probabilistic search designs, including those that encompass probabilistic rules that have been highly favored in previous SAT approaches. These outcomes motivate a closer look at surrogate strategies and more advanced ways of integrating them with adaptive memory and learning procedures.

Parallel image restoration using surrogate constraint methods

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.

Dittman and Staats give an algorithm to solve the linear programming problem by a surrogate constraint method. This technical note illustrates that the proposed algorithm will only converge in an infinite number of steps to the optimal solution of the original linear programming problem. This shows a serious drawback of the algorithm. A remedial modification is suggested which will lead to problem solution.

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 the 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. We present a scatter-search implementation designed to find high-quality solutions for the 3D image-registration problem, which has many practical applications. This problem arises in computer vision applications when finding a correspondence or transformation between two computer images obtained under different conditions. Our implementation goes beyond a simple exercise on applying scatter search, by incorporating innovative mechanisms to combine and improve solutions and to create a balance between intensification and diversification in the reference set. Furthermore, heuristic information taken from a preprocessing of the images is incorporated into the algorithm to improve its performance. Our computational experimentation tackling two different medical registration applications establishes the effectiveness of scatter search in relation to different approaches usually applied to solving the problem. We have considered both simulated magnetic resonance images and real-world computerized tomography images as data sets. To measure the robustness of our proposal, the image data sets are intentionally selected for addressing registration environments with the presence of noise, anatomical lesions, and occlusions between images.

Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

Zero duality gap in integer programming: P-norm surrogate constraint method

Multidimensional multiextremal optimization problems and numerical methods for solving them are studied. The objective function is supposed to satisfy the Lipschitz condition with an a priori unknown constant, which is the only general assumption imposed on it. Problems of this type often arise in applications. Two dimensionality reduction approaches to multidimensional optimization problems, i.e., the use of Peano curves (evolvents) and a recursive multistep scheme, are considered. A generalized scheme combining both approaches is proposed. In the new scheme, an original multidimensional problem is reduced to a family of lower-dimensional problems, which are solved using evolvents. An adaptive algorithm with the simultaneous solution of all resulting subproblems is implemented. Computational experiments on several hundred test problems are performed. In accordance with experimental evidence, the new dimensional reduction scheme is effective.

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems.

The paper claims that the primary importance in solving the classification problem is to find the conditions for dividing the General complexity into classes, determine the quality of such a bundle, and verify the classifier model. We consider a mathematical model of a non-randomized classifier of features obtained without a teacher, when the number of classes is not set a priori, but only its upper bound is set. The mathematical model is presented in the form of a statement of a minimax conditional extreme task, and it is a problem of searching for the matrix of belonging of objects to a class, and representative (reference) elements within each class. The development of the feature classifier is based on the synthesis of two-dimensional probability density in the coordinate space: classes-objects. Using generalized functions, the probabilistic problem of finding the minimum Bayesian risk is reduced to a deterministic problem on a set of non-randomized classifiers. At the same time, the use of specially introduced constraints fixes non-randomized decision rules and plunges the integer problem of nonlinear programming into a General continuous nonlinear problem. For correct synthesis of the classifier, the dispersion curve of the isotropic sample is necessary. It is necessary to use the total intra-class and inter-class variance to characterize the quality of classification. The classification problem can be interpreted as a particular problem of the theory of catastrophes. Under the conditions of limited initial data, a minimax functional was found that reflects the quality of classification for a quadratic loss function. The developed mathematical model is classified as an integer nonlinear programming problem. The model is given using polynomial constraints to the form of a General problem of nonlinear continuous programming. The necessary conditions for the bundle into classes are found. These conditions can be used as sufficient when testing the hypothesis about the existence of classes.

Software-defined networking (SDN) allows the smart grid to be centrally controlled and managed by decoupling the control plane from the data plane, but it also expands attack surface for attackers. Existing studies about the security of SDN-enabled smart grid (SDSG) mainly focused on static methods such as access control and identity authentication, which is vulnerable to attackers that carefully probe the system. As the attacks become more variable and complex, there is an urgent need for dynamic defense methods. In this paper, we propose a security function virtualization (SFV) based moving target defense of SDSG which makes the attack surface constantly changing. First, we design a dynamic defense mechanism by migrating virtual security function (VSF) instances as the traffic state changes. The centralized SDN controller is re-designed for global status monitoring and migration management. Moreover, we formalize the VSF instances migration problem as an integer nonlinear programming problem with multiple constraints and design a pre-migration algorithm to prevent VSF instances' resources from being exhausted. Simulation results indicate the feasibility of the proposed scheme.

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice.

In this study, we introduce a new global optimization technique for a multi-dimensional unconstrained optimization problem. First, we present a new smoothing auxiliary function. Second, we transform the multi-dimensional problem into a one-dimensional problem by using an auxiliary function to reduce the number of local minimizers and then we find the global minimizer of the one-dimensional problem. Finally, we find the global minimizer of the multi-dimensional smooth objective function with the help of a new algorithm.

Extension of an exponential discretization scheme to multidimensional convection–diffusion problems