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On relaxing the Mangasarian-Fromovitz constraint qualification

- Kruger, Alexander, Minchenko, Leonld, Outrata, Jiri

**Authors:**Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Positivity Vol. 18, no. 1 (2014), p. 171-189**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.

**Authors:**Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Positivity Vol. 18, no. 1 (2014), p. 171-189**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.

On topological existence theorems and applications to optimization-related problems

- Khanh, Phan Quoc, Lin, Lai Jiu, Long, Vo Si Trong

**Authors:**Khanh, Phan Quoc , Lin, Lai Jiu , Long, Vo Si Trong**Date:**2014**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 79, no. 3 (June 2014 2014), p. 253-272**Full Text:**false**Reviewed:****Description:**In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.**Description:**C1

Optimality conditions and optimization methods for quartic polynomial optimization

- Wu, Zhiyou, Tian, Jing, Quan, Jing, Ugon, Julien

**Authors:**Wu, Zhiyou , Tian, Jing , Quan, Jing , Ugon, Julien**Date:**2014**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 232, no. (2014), p. 968-982**Full Text:**false**Reviewed:****Description:**In this paper multivariate quartic polynomial optimization program (QPOP) is considered. Quartic optimization problems arise in various practical applications and are proved to be NP hard. We discuss necessary global optimality conditions for quartic problem (QPOP). And then we present a new (strongly or ε-strongly) local optimization method according to necessary global optimality conditions, which may escape and improve some KKT points. Finally we design a global optimization method for problem (QPOP) by combining the new (strongly or ε-strongly) local optimization method and an auxiliary function. Numerical examples show that our algorithms are efficient and stable.

Post-buckling solutions of hyper-elastic beam by canonical dual finite element method

- Cai, Kun, Gao, David, Qin, Qing

**Authors:**Cai, Kun , Gao, David , Qin, Qing**Date:**2014**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Solids Vol. 19, no. 6 (2014), p. 659-671**Full Text:**false**Reviewed:****Description:**The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.

- Bagirov, Adil, Miettinen, Kaisa, Weber, Gerhard-Wilhelm

**Authors:**Bagirov, Adil , Miettinen, Kaisa , Weber, Gerhard-Wilhelm**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 60, no. 1 (June 2014), p. 1-3**Full Text:**false**Reviewed:****Description:**C1

- Beremlijski, Petr, Haslinger, Jaroslav, Outrata, Jiri, Pathó, Róbert

**Authors:**Beremlijski, Petr , Haslinger, Jaroslav , Outrata, Jiri , Pathó, Róbert**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 5 (2014), p. 3371-3400**Full Text:**false**Reviewed:****Description:**The present paper deals with shape optimization in discretized two-dimensional (2D) contact problems with Coulomb friction, where the coefficient of friction is assumed to depend on the unknown solution. Discretization of the continuous state problem leads to a system of finite-dimensional implicit variational inequalities, parametrized by the so-called design variable, that determines the shape of the underlying domain. It is shown that if the coefficient of friction is Lipschitz and sufficiently small in the C0,1 -norm, then the discrete state problems are uniquely solvable for all admissible values of the design variable (the admissible set is assumed to be compact), and the state variables are Lipschitzian functions of the design variable. This facilitates the numerical solution of the discretized shape optimization problem by the so-called implicit programming approach. Our main results concern sensitivity analysis, which is based on the well-developed generalized differential calculus of B. Mordukhovich and generalizes some of the results obtained in this context so far. The derived subgradient information is then combined with the bundle trust method to compute several model examples, demonstrating the applicability and efficiency of the presented approach. © 2014 Society for Industrial and Applied Mathematics

Sigma supporting cone and optimality conditions in non-convex problems

- Hassani, Sara, Mammadov, Musa

**Authors:**Hassani, Sara , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Far East Journal of Mathematical Sciences Vol. 91, no. 2 (2014), p. 169-190**Full Text:**false**Reviewed:****Description:**In this paper, a new supporting function for characterizing non-convex sets is introduced. The notions of Ïƒ-supporting cone and maximal conic gap are proposed and some properties are investigated. By applying these new notions, we establish the optimality conditions considered in [7] for a broader class of finite dimensional normed spaces in terms of weak subdifferentials.

Solving second-order conic systems with variable precision

- Cucker, Felipe, Peña, Javier, Roshchina, Vera

**Authors:**Cucker, Felipe , Peña, Javier , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 150, no. 2 (2014), p. 217-250**Full Text:**false**Reviewed:****Description:**We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited. © 2014, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Some preconditioners for systems of linear inequalities

- Peña, Javier, Roshchina, Vera, Soheili, Negar

**Authors:**Peña, Javier , Roshchina, Vera , Soheili, Negar**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 8, no. 7 (2014), p. 2145-2152**Full Text:**false**Reviewed:****Description:**We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities. © 2014, Springer-Verlag Berlin Heidelberg.

Special Issue on recent advances in continuous optimization on the occasion of the 25th European conference on Operational Research (EURO XXV 2012)

- Weber, Gerhard-Wilhelm, Kruger, Alexander, Martinez-Legaz, Juan, Mordukhovich, Boris, Sakalauskas, Leonidas

**Authors:**Weber, Gerhard-Wilhelm , Kruger, Alexander , Martinez-Legaz, Juan , Mordukhovich, Boris , Sakalauskas, Leonidas**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Vol. 63, no. 1 (2014), p. 1-5**Full Text:****Reviewed:**

**Authors:**Weber, Gerhard-Wilhelm , Kruger, Alexander , Martinez-Legaz, Juan , Mordukhovich, Boris , Sakalauskas, Leonidas**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Vol. 63, no. 1 (2014), p. 1-5**Full Text:****Reviewed:**

Spline regression models for complex multi-modal regulatory networks

- Ozmen, Ayse, Kropat, Erik, Weber, Gerhard-Wilhelm

**Authors:**Ozmen, Ayse , Kropat, Erik , Weber, Gerhard-Wilhelm**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 29, no. 3 (2014), p. 515-534**Full Text:**false**Reviewed:****Description:**Complex regulatory networks often have to be further expanded and improved with regard to the unknown effects of additional parameters and factors that can emit a disturbing influence on the key variables under consideration. The concept of target-environment (TE) networks provides a holistic framework for the analysis of such parameter-dependent multi-modal systems. In this study, we consider time-discrete TE regulatory systems with spline entries. We introduce a new regression model for these particular two-modal systems that allows us to determine the unknown system parameters by applying the multivariate adaptive regression spline (MARS) technique and the newly developed conic multivariate adaptive regression spline (CMARS) method. We obtain a relaxation by means of continuous optimization, especially, conic quadratic programming (CQP) that could be conducted by interior point methods. Finally, a numerical example demonstrates the efficiency of the spline-based approach.

Structure learning of Bayesian Networks using global optimization with applications in data classification

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

Turnpike theorem for an infinite horizon optimal control problem with time delay

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

**Authors:**Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Control and Optimization Vol. 52, no. 1 (2014), p. 420-438**Full Text:****Reviewed:****Description:**An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model. Â© 2014 Society for Industrial and Applied Mathematics.

- Jelinek, Herbert, Kelarev, Andrei, Robinson, Dean, Stranieri, Andrew, Cornforth, David

**Authors:**Jelinek, Herbert , Kelarev, Andrei , Robinson, Dean , Stranieri, Andrew , Cornforth, David**Date:**2014**Type:**Text , Journal article**Relation:**Applied Soft Computing Vol. 14, no. PART A (2014), p. 81-87**Full Text:**false**Reviewed:****Description:**This work investigates the effectiveness of using computer-based machine learning regression algorithms and meta-regression methods to predict performance data for Australian football players based on parameters collected during daily physiological tests. Three experiments are described. The first uses all available data with a variety of regression techniques. The second uses a subset of features selected from the available data using the Random Forest method. The third used meta-regression with the selected feature subset. Our experiments demonstrate that feature selection and meta-regression methods improve the accuracy of predictions for match performance of Australian football players based on daily data of medical tests, compared to regression methods alone. Meta-regression methods and feature selection were able to obtain performance prediction outcomes with significant correlation coefficients. The best results were obtained by the additive regression based on isotonic regression for a set of most influential features selected by Random Forest. This model was able to predict athlete performance data with a correlation coefficient of 0.86 (p < 0.05). Â© 2013 Published by Elsevier B.V. All rights reserved.**Description:**C1

- Zhou, Xiaojun, Dong, Tianxue, Tang, Xiaolin, Yang, Chunhua, Gui, Weihua

**Authors:**Zhou, Xiaojun , Dong, Tianxue , Tang, Xiaolin , Yang, Chunhua , Gui, Weihua**Date:**2015**Type:**Text , Journal article**Relation:**Optimal Control Applications and Methods Vol. 36, no. 6 (2015), p. 844-852**Full Text:**false**Reviewed:****Description:**In this study, the guaranteed cost control of discrete time uncertain system with both state and input delays is considered. Sufficient conditions for the existence of a memoryless state feedback guaranteed cost control law are given in the bilinear matrix inequality form, which needs much less auxiliary matrix variables and storage space. Furthermore, the design of guaranteed cost controller is reformulated as an optimization problem with a linear objective function, bilinear, and linear matrix inequalities constraints. A nonlinear semi-definite optimization solver - PENLAB is used as a solution technique. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

A heuristic algorithm for solving the minimum sum-of-squares clustering problems

**Authors:**Ordin, Burak , Bagirov, Adil**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 61, no. 2 (2015), p. 341-361**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.

A reliability-based design optimization model for electricity power networks

**Authors:**Ezzati, Ghasem**Date:**2015**Type:**Text , Journal article**Relation:**Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 22, no. 5 (2015), p. 339-357**Full Text:**false**Reviewed:****Description:**Significant attentions have recently been attracted by electricity power net- works where many optimization models are applied to optimize distributed power. Many optimization models are available for electricity networks that mainly take into accoun- t total cost. Reliability related issues of electricity networks are also considered in the literature. However, there is a lack to formulate a reliability-based design optimization (RBDO) model of these networks. An RBDO model is introduced in this paper to deal with probabilistic constraints in an optimization model for electricity networks. In our suggested approach, an optimization problem is firstly solved to find optimal parameters of the network. Then, the optimal solution is adjusted using an RBDO problem. Our main aim is to minimize an extra cost that is experienced by considering reliability. It is expected to have a higher extra cost for a lower failure probability. © 2015 Watam Press.

An incremental clustering algorithm based on hyperbolic smoothing

- Bagirov, Adil, Ordin, Burak, Ozturk, Gurkan, Xavier, Adilson

**Authors:**Bagirov, Adil , Ordin, Burak , Ozturk, Gurkan , Xavier, Adilson**Date:**2015**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 61, no. 1 (2015), p. 219-241**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is an important problem in data mining. It can be formulated as a nonsmooth, nonconvex optimization problem. For the most global optimization techniques this problem is challenging even in medium size data sets. In this paper, we propose an approach that allows one to apply local methods of smooth optimization to solve the clustering problems. We apply an incremental approach to generate starting points for cluster centers which enables us to deal with nonconvexity of the problem. The hyperbolic smoothing technique is applied to handle nonsmoothness of the clustering problems and to make it possible application of smooth optimization algorithms to solve them. Results of numerical experiments with eleven real-world data sets and the comparison with state-of-the-art incremental clustering algorithms demonstrate that the smooth optimization algorithms in combination with the incremental approach are powerful alternative to existing clustering algorithms.

An induction theorem and nonlinear regularity models

- Khanh, Phan, Kruger, Alexander, Thao, Nguyen

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

Canonical duality theory and triality for solving general global optimization problems in complex systems

- Morales-Silva, Daniel, Gao, David

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

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