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28Bagirov, Adil
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On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

Perturbation of error bounds

- Kruger, Alexander, López, Marco, Théra, Michel

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

**Authors:**Kruger, Alexander , López, Marco , Théra, Michel**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Set regularities and feasibility problems

- Kruger, Alexander, Luke, Russell, Thao, Nguyen

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2018**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

Strong metric subregularity of mappings in variational analysis and optimization

- Cibulka, Radek, Dontchev, Asen, Kruger, Alexander

**Authors:**Cibulka, Radek , Dontchev, Asen , Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1247-1287**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form f+F, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.

A comparison of bidding strategies for online auctions using fuzzy reasoning and negotiation decision functions

- Kaur, Preetinder, Goyal, Madhu, Lu, Jie

**Authors:**Kaur, Preetinder , Goyal, Madhu , Lu, Jie**Date:**2017**Type:**Text , Journal article**Relation:**IEEE Transactions on Fuzzy Systems Vol. 25, no. 2 (2017), p. 425-438**Full Text:****Reviewed:****Description:**Bidders often feel challenged when looking for the best bidding strategies to excel in the competitive environment of multiple and simultaneous online auctions for same or similar items. Bidders face complicated issues for deciding which auction to participate in, whether to bid early or late, and how much to bid. In this paper, we present the design of bidding strategies, which aim to forecast the bid amounts for buyers at a particular moment in time based on their bidding behavior and their valuation of an auctioned item. The agent develops a comprehensive methodology for final price estimation, which designs bidding strategies to address buyers' different bidding behaviors using two approaches: Mamdani method with regression analysis and negotiation decision functions. The experimental results show that the agents who follow fuzzy reasoning with a regression approach outperform other existing agents in most settings in terms of their success rate and expected utility.

**Authors:**Kaur, Preetinder , Goyal, Madhu , Lu, Jie**Date:**2017**Type:**Text , Journal article**Relation:**IEEE Transactions on Fuzzy Systems Vol. 25, no. 2 (2017), p. 425-438**Full Text:****Reviewed:****Description:**Bidders often feel challenged when looking for the best bidding strategies to excel in the competitive environment of multiple and simultaneous online auctions for same or similar items. Bidders face complicated issues for deciding which auction to participate in, whether to bid early or late, and how much to bid. In this paper, we present the design of bidding strategies, which aim to forecast the bid amounts for buyers at a particular moment in time based on their bidding behavior and their valuation of an auctioned item. The agent develops a comprehensive methodology for final price estimation, which designs bidding strategies to address buyers' different bidding behaviors using two approaches: Mamdani method with regression analysis and negotiation decision functions. The experimental results show that the agents who follow fuzzy reasoning with a regression approach outperform other existing agents in most settings in terms of their success rate and expected utility.

A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes

- Joki, Kaisa, Bagirov, Adil, Karmitsa, Napsu, Makela, Marko

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 68, no. 3 (2017), p. 501-535**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an -critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.

A scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

- Natarajan, Sundararajan, Ooi, Ean Tat, Saputra, Albert, Song, Chongmin

**Authors:**Natarajan, Sundararajan , Ooi, Ean Tat , Saputra, Albert , Song, Chongmin**Date:**2017**Type:**Text , Journal article**Relation:**Engineering Analysis with Boundary Elements Vol. 80, no. (2017), p. 218-229**Full Text:**false**Reviewed:****Description:**In this paper, a displacement based finite element framework for general three-dimensional convex polyhedra is presented. The method is based on a semi-analytical framework, the scaled boundary finite element method. The method relies on the definition of a scaling center from which the entire boundary is visible. The salient feature of the method is that the discretizations are restricted to the surfaces of the polyhedron, thus reducing the dimensionality of the problem by one. Hence, an explicit form of the shape functions inside the polyhedron is not required. Conforming shape functions defined over arbitrary polygon, such as the Wachpress interpolants are used over each surface of the polyhedron. Analytical integration is employed within the polyhedron. The proposed method passes patch test to machine precision. The convergence and the accuracy properties of the method is discussed by solving few benchmark problems in linear elasticity. © 2017 Elsevier Ltd

A unifying approach to robust convex infinite optimization duality

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.

Borwein–Preiss vector variational principle

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails

- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien

**Authors:**Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964**Full Text:****Reviewed:****Description:**In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.

**Authors:**Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964**Full Text:****Reviewed:****Description:**In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.

DC programming algorithm for clusterwise linear L1 regression

**Authors:**Bagirov, Adil , Taheri, Sona**Date:**2017**Type:**Text , Journal article**Relation:**Journal of the Operations Research Society of China Vol. 5, no. 2 (2017), p. 233-256**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The aim of this paper is to develop an algorithm for solving the clusterwise linear least absolute deviations regression problem. This problem is formulated as a nonsmooth nonconvex optimization problem, and the objective function is represented as a difference of convex functions. Optimality conditions are derived by using this representation. An algorithm is designed based on the difference of convex representation and an incremental approach. The proposed algorithm is tested using small to large artificial and real-world data sets. © 2017, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg.

Density character of subgroups of topological groups

- Leiderman, Arkady, Morris, Sidney, Tkachenko, Mikhail

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Double well potential function and its optimization in the n-dimensional real space - Part I

- Fang, Shucherng, Gao, David, Lin, Gang-Xuan, Sheu, Ruey-Lin, Xing, Wenxun

**Authors:**Fang, Shucherng , Gao, David , Lin, Gang-Xuan , Sheu, Ruey-Lin , Xing, Wenxun**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 13, no. 3 (2017), p. 1291-1305**Full Text:**false**Reviewed:****Description:**A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. It is derived from a discrete approx imation of the generalized Ginzburg-Landau functional, and we are interested in understanding its global minimum solution and all local non-global points. The main difficulty for the model is due to its non-convexity. In Part I of the paper, we first characterize the global minimum solution set, whereas the study for local non-global optimal solutions is left for Part II. We show that, the dual of the Lagrange dual of the double well potential problem is a linearly constrained convex minimization problem, which, under a designated nonlin ear transformation, can be equivalently mapped to a portion of the original double well potential function containing the global minimum. In other words, solving the global minimum of the double well potential function is essentially a convex minimization problem, despite of its non-convex nature. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.

Double well potential function and its optimization in the N-dimensional real space -- Part I

- Fang, Shucherng, Gao, David, Lin, Gang-Xuan, Sheu, Ruey-Lin, Xing, Wenxun

**Authors:**Fang, Shucherng , Gao, David , Lin, Gang-Xuan , Sheu, Ruey-Lin , Xing, Wenxun**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 13, no. 3 (2017), p. 1291-1305**Full Text:**false**Reviewed:****Description:**A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. It is derived from a discrete approximation of the generalized Ginzburg-Landau functional, and we are interested in understanding its global minimum solution and all local non-global points. The main difficulty for the model is due to its non-convexity. In Part I of the paper, we first characterize the global minimum solution set, whereas the study for local non-global optimal solutions is left for Part II. We show that, the dual of the Lagrange dual of the double well potential problem is a linearly constrained convex minimization problem, which, under a designated nonlinear transformation, can be equivalently mapped to a portion of the original double well potential function containing the global minimum. In other words, solving the global minimum of the double well potential function is essentially a convex minimization problem, despite of its non-convex nature. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.

Farkas-type results for vector-valued functions with applications

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Mo, T. H.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390**Full Text:****Reviewed:****Description:**The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390**Full Text:****Reviewed:****Description:**The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

On modeling and global solutions for d.c. optimization problems by canonical duality theory

**Authors:**Jin, Zhong , Gao, David**Date:**2017**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 296, no. (2017), p. 168-181**Full Text:****Reviewed:****Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.**Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented. © 2016 Elsevier Inc.

**Authors:**Jin, Zhong , Gao, David**Date:**2017**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 296, no. (2017), p. 168-181**Full Text:****Reviewed:****Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.**Description:**This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented. © 2016 Elsevier Inc.

On the Aubin property of a class of parameterized variational systems

- Gfrerer, Helmut, Outrata, Jiri

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2017**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 86, no. 3 (2017), p. 443-467**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples. © 2017, Springer-Verlag GmbH Germany.

**Authors:**Gfrerer, Helmut , Outrata, Jiri**Date:**2017**Type:**Text , Journal article**Relation:**Mathematical Methods of Operations Research Vol. 86, no. 3 (2017), p. 443-467**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples. © 2017, Springer-Verlag GmbH Germany.

On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor

- Gao, David, Neff, Patrizio, Roventa, Ionel, Thiel, Christian

**Authors:**Gao, David , Neff, Patrizio , Roventa, Ionel , Thiel, Christian**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Elasticity Vol. 127, no. 2 (2017), p. 303-308**Full Text:****Reviewed:****Description:**We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies which are convex with respect to the right Cauchy-Green tensor , where denotes the gradient of deformation. Examples of such energies exhibiting a blow up for are given.

**Authors:**Gao, David , Neff, Patrizio , Roventa, Ionel , Thiel, Christian**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Elasticity Vol. 127, no. 2 (2017), p. 303-308**Full Text:****Reviewed:****Description:**We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies which are convex with respect to the right Cauchy-Green tensor , where denotes the gradient of deformation. Examples of such energies exhibiting a blow up for are given.

- Wu, Zhiyou, Bai, Fusheng, Tian, Jing

**Authors:**Wu, Zhiyou , Bai, Fusheng , Tian, Jing**Date:**2017**Type:**Text , Journal article**Relation:**Journal of the Operations Research Society of China Vol. 5, no. 2 (2017), p. 193-218**Full Text:**false**Reviewed:****Description:**In this paper, an optimality condition for nonlinear programming problems with box constraints is given by using linear transformation and Lagrange interpolating polynomials. Based on this condition, two new local optimization methods are developed. The solution points obtained by the new local optimization methods can improve the Karush–Kuhn–Tucker (KKT) points in general. Two global optimization methods then are proposed by combining the two new local optimization methods with a filled function method. Some numerical examples are reported to show the effectiveness of the proposed methods. © 2017, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg.

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