Weak stationarity : Eliminating the gap between necessary and sufficient conditions
- Authors: Kruger, Alexander
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 2 (Apr 2004), p. 147-164
- Full Text:
- Reviewed:
- Description: Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity).
- Description: 2003000887
On global optimality conditions via separation functions
- Authors: Rubinov, Alex , Uderzo, A.
- Date: 2001
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 109, no. 2 (May 2001), p. 345-370
- Full Text: false
- Reviewed:
- Description: The paper examines some axiomatic definitions of separation functions that can be employed fruitfully in the analysis of side-constrained extremum problems. A study of their general properties points out connections with abstract convex analysis and recent generalizations of Lagrangian approaches to duality and exact penalty methods. Many concrete examples are brought out.
Global optimality conditions for some classes of optimization problems
- Authors: Wu, Zhiyou , Rubinov, Alex
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 145, no. 1 (2009), p. 164-185
- Full Text: false
- Reviewed:
- Description: We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity. © 2009 Springer Science+Business Media, LLC.
Nonsmooth analysis : Fréchet subdifferentials
- Authors: Kruger, Alexander
- Date: 2009
- Type: Text , Book chapter
- Relation: Encyclopedia of Optimization Chapter p. 2651-2658
- Full Text: false
On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions
- Authors: Kasimbeyli, Refail , Mammadov, Musa
- Date: 2009
- Type: Text , Journal article
- Relation: Siam Journal on Optimization Vol. 20, no. 2 (2009), p. 841-855
- Full Text:
- Reviewed:
- Description: In this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions.
Second-order variational analysis in conic programming with applications to optimality and stability
- Authors: Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector
- Date: 2015
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text:
- Reviewed:
- Description: This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.
Sigma supporting cone and optimality conditions in non-convex problems
- 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 σ-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.
Sufficient global optimality conditions for weakly convex minimization problems
- Authors: Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 39, no. 3 (2007), p. 427-440
- Full Text: false
- Reviewed:
- Description: In this paper, we present sufficient global optimality conditions for weakly convex minimization problems using abstract convex analysis theory. By introducing (L,X)-subdifferentials of weakly convex functions using a class of quadratic functions, we first obtain some sufficient conditions for global optimization problems with weakly convex objective functions and weakly convex inequality and equality constraints. Some sufficient optimality conditions for problems with additional box constraints and bivalent constraints are then derived. © 2007 Springer Science+Business Media, Inc.
- Description: C1
- Description: 2003005516
On regular coderivatives in parametric equilibria with non-unique multipliers
- Authors: Henrion, René , Outrata, Jiri , Surowiec, Thomas
- Date: 2012
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 136, no. 1 (December 2012), p. 111-131
- Full Text: false
- Reviewed:
- Description: This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations (GEs). The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such GEs. The advantages are illustrated by means of examples.
- Description: C1
Solving facility location problem based on duality approach
- Authors: Ruan, Ning
- Date: 2015
- Type: Text , Conference paper
- Relation: 3rd World Congress on Global Optimization in Engineering and Science, WCGO 2013; Anhui, China; 8th-12th July 2013 Vol. 95, p. 165-172
- Full Text: false
- Reviewed:
- Description: The facility location problem is one of the most widely studied discrete location problems, whose applications arise in a variety of settings, such as routers or servers in a communication network, warehouses or distribution centres in a supply chain, hospitals or airports in a public service system. The problem involves locating a number of facilities to minimize the sum of the fixed setup costs and the variable costs of serving the market demand from these facilities. First a dual problem is developed for the facility location problem. Then general optimality conditions are also obtained, which generate sequences globally converging to a primal and dual solutions, respectively. © Springer International Publishing Switzerland 2015.
Optimality conditions for nonsmooth optimization problems via generalised derivatives
- Authors: Hassani, Sara
- Date: 2016
- Type: Text , Thesis , PhD
- Full Text:
- Description: Aquatic plants are integral components of freshwater ecosystems and provide essential ecosystem services. However, when invasive species establish in new aquatic environments, there are few natural checks and balances to inhibit their growth and spread. Overabundant aquatic vegetation can harm aquatic systems if left unchecked and negatively impact on agricultural productivity, social amenity and biodiversity values. Prevention and early intervention are recognised as the most cost effective means to manage invasive species that pose a biosecurity risk. This thesis contributes to the development of effective management strategies for one of the world’s most invasive aquatic plant species, known as alligator weed (Alternanthera philoxeroides (Mart.) Griseb.). It focusses on developing management strategies in an early stage of invasion, in order to achieve extirpation of this species from catchments and waterways. Developing effective detection and surveillance strategies are required for invasive aquatic plants, as a key impediment to achieving extirpation is the ability to detect infestations, so that control strategies can be enacted. This thesis investigates the effectiveness of aerial surveillance for detection of alligator weed at different spatial scales, using high altitude aerial imagery (orthophotos) and unmanned aerial vehicle (UAV) technology. An examination of the growth rate of alligator weed in Victoria, Australia, over a five year period, demonstrates the effective use of orthophotos to detect and monitor large infestations of aquatic alligator weed. The efficacy of unmanned aerial vehicle technology, including the use of automated algorithms, to detect patches of alligator weed growing in waterways is evaluated against current detection techniques. Effective management of invasive aquatic plants targeted for extirpation requires the coupling of effective detection and control efforts to prevent reproduction. To date, development of control strategies for aquatic alligator weed has been limited to evaluating the efficacy of short-term control at a local scale without regard to the effects of management strategies on dispersal of propagules throughout catchments. This thesis determines that viable alligator weed stem fragments are produced following herbicide application, which comprises extirpation efforts. This thesis has gone further than current practice in that it has evaluated the efficacy of current and novel control techniques, in both laboratory and field trials and has developed methods to manage viable fragment production post-herbicide application, to limit dispersal throughout catchments. In this respect, the application of the herbicides glyphosate, metsulfuron-methyl and imazapyr, and their effectiveness when incorporating surfactant systems and plant growth regulators, have been evaluated in field and laboratory studies to optimise control techniques for aquatic alligator weed. Results have shown that our approaches, when used in an early stage of invasion, are capable of eliminating patches of alligator weed in two to three years. Integral to the research is an experiment to determine the effect of herbicide treatments on the production of alligator weed stem fragments and their subsequent viability. Further investigation to determine the usefulness of commercially available plant growth regulators (PGRs) to reduce the number of viable propagules produced by alligator weed post-herbicide application was found to be ineffective. This thesis also evaluates the impact of herbicides and surfactant systems, on all key alligator weed response metrics in aquatic environments including; above ground biomass, below ground biomass and viable stem fragmentation. No previous studies have looked simultaneously at these three important measures for determining the efficacy of a particular control regime, and we have determined that this is essential for effective management of aquatic alligator weed in an early stage of invasion. The thesis has underscored the notion that development of more effective management strategies, based upon experimental trials, will result in an increased likelihood of eradicating invasive aquatic plants that pose a biosecurity risk, and thus move toward the mitigation of the threat that high-risk species pose to aquatic ecosystems. PLEASE NOTE: Portions of the full text have been removed due to copyright restrictions.
- Description: Doctor of Philosophy
An induction theorem and nonlinear regularity models
- 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.
Optimization methods for box-constrained nonlinear programming problems based on linear transformation and Lagrange interpolating polynomials
- 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.
Enlargements of the moreau–rockafellar subdifferential
- Authors: Abbasi, Malek , Kruger, Alexander , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text:
- Reviewed:
- Description: This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup
Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus
- Authors: Correa, R. , Hantoute, A. , López, Marco
- Date: 2021
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.
Subdifferential of the supremum via compactification of the index set
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2020
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 48, no. 3 (2020), p. 569-588, http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
- Description: Funding details: Fondo Nacional de Desarrollo CientÃfico, Tecnológico y de Innovación Tecnológica, FONDECYT, PIA AFB-170001, 1190110, 1190012 Funding details: Universidad de Alicante, BEA- GAL 18/00205, PGC2018-097960-B-C21 Funding details: Australian Research Council, ARC, DP 180100602 Funding details: Comisión Nacional de Investigación CientÃfica y Tecnológica, CONICYT Funding details: Ministerio de Ciencia e Innovación, MICINN Funding text 1: Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEA- GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602
Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness
- Authors: Kruger, Alexander , Mehlitz, Patrick
- Date: 2022
- Type: Text , Journal article
- Relation: ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text:
- Reviewed:
- Description: Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors.