Search ResultsShowing 1-18 of 18
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arXiv:2501.04711 (Published 2024-12-29)
Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions
Comments: arXiv admin note: substantial text overlap with arXiv:2409.19636Categories: math.OCIn this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function.
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arXiv:2312.10086 (Published 2023-12-12)
A Necessary Optimality Condition for Extended Weighted Generalized Fractional Optimal Control Problems
Comments: This is a preprint version of the paper published open access in 'Results in Control and Optimization 14 (2024), Art. 100356, 5 pp' [https://doi.org/10.1016/j.rico.2023.100356]Journal: Results in Control and Optimization 14 (2024), Art. 100356, 5 ppCategories: math.OCKeywords: generalized fractional optimal control problems, weighted generalized fractional optimal control, extended weighted generalized fractional optimal, necessary optimality condition, atangana-baleanu fractional dynamic optimizationTags: journal articleUsing the recent weighted generalized fractional order operators of Hattaf, a general fractional optimal control problem without constraints on the values of the control functions is formulated and a corresponding (weak) version of Pontryagin's maximum principle is proved. As corollaries, necessary optimality conditions for Caputo-Fabrizio, Atangana-Baleanu and weighted Atangana-Baleanu fractional dynamic optimization problems are trivially obtained. As an application, the weighted generalized fractional problem of the calculus of variations is investigated and a new more general fractional Euler-Lagrange equation is given.
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arXiv:2104.13244 (Published 2021-04-27)
A Unifying Framework for Sparsity Constrained Optimization
Categories: math.OCIn this paper, we consider the optimization problem of minimizing a continuously differentiable function subject to both convex constraints and sparsity constraints. By exploiting a mixed-integer reformulation from the literature, we define a necessary optimality condition based on a tailored neighborhood that allows to take into account potential changes of the support set. We then propose an algorithmic framework to tackle the considered class of problems and prove its convergence to points satisfying the newly introduced concept of stationarity. We further show that, by suitably choosing the neighborhood, other well-known optimality conditions from the literature can be recovered at the limit points of the sequence produced by the algorithm. Finally, we analyze the computational impact of the neighborhood size within our framework and in the comparison with some state-of-the-art algorithms, namely, the Penalty Decomposition method and the Greedy Sparse-Simplex method. The algorithms have been tested using a benchmark related to sparse logistic regression problems.
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arXiv:1901.01507 (Published 2019-01-06)
Some remarks on weak generalizations of minima and quasi efficiency
Comments: 9 pages, forum paperCategories: math.OCIn this note, we remark, with sufficient mathematical rigor, that many weak generalizations of the usual minimum available in the literature are not true generalizations. Motivated by the Ekeland Variational Principle, we provide, first time, the criteria for weaker generalizations of the usual minimum. Further, we show that the quasi efficiency, recently used in Bhatia et al. (Optim. Lett. 7, 127-135 (2013)) and introduced in Gupta et al. ( Bull. Aust. Math. Soc. 74, 207-218 (2006)) is not a true generalization of the usual efficiency. Since the former paper relies heavily on the results of later one, so we discuss the later paper. We show that the necessary optimality condition is a consequence of the local Lipschitzness and sufficiency result is trivial in the later paper. Consequently, the duality results of the same paper are also inconsistent.
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arXiv:1810.12119 (Published 2018-10-26)
A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier-Stokes Equations
Comments: arXiv admin note: text overlap with arXiv:1809.00911Categories: math.OCWe consider the control problem of the stochastic Navier-Stokes equations in multidimensional domains introduced in \cite{ocpc} restricted to noise terms defined by Q-Wiener processes. Using a stochastic maximum principle, we derive a necessary optimality condition to design the optimal control based on an adjoint equation, which is given by a backward SPDE. Moreover, we show that the optimal control satisfies a sufficient optimality condition. As a consequence, we can solve uniquely control problems constrained by the stochastic Navier-Stokes equations especially for two-dimensional as well as for three-dimensional domains.
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arXiv:1607.00809 (Published 2016-07-04)
Optimal control of a rate-independent evolution equation via viscous regularization
We study the optimal control of a rate-independent system that is driven by a convex, quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.
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arXiv:1506.08932 (Published 2015-06-30)
Optimal Control of Continuity Equations
Categories: math.OCAn optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.
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arXiv:1403.3937 (Published 2014-03-16)
Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition --- Application to fractional variational problems
Comments: This is a preprint of a paper whose final and definite form will appear in Differential and Integral Equations, ISSN 0893-4983 (See http://www.aftabi.com/DIE.html). Submitted 19/July/2013; Accepted 16/March/2014Journal: Differential Integral Equations 27 (2014), no. 7/8, 743--766Categories: math.OCKeywords: necessary optimality condition, generalized lagrangian functionals, fractional variational problems, application, study dynamic minimization problemsTags: journal articleWe study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. Main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].
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arXiv:1303.4075 (Published 2013-03-17)
Noether's theorem for fractional variational problems of variable order
Comments: This is a preprint of a paper whose final and definite form will appear in Central European Journal of Physics. Paper submitted 30-Jan-2013; revised 12-March-2013; accepted for publication 15-March-2013Journal: Cent. Eur. J. Phys. 11 (2013), no. 6, 691--701Keywords: fractional variational problems, noethers theorem, necessary optimality condition, euler-lagrange type, derivativesTags: journal articleWe prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.
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arXiv:1209.1530 (Published 2012-09-07)
Hahn's Symmetric Quantum Variational Calculus
Comments: This is a preprint of a paper whose final and definite form will appear in the international journal Numerical Algebra, Control and Optimization (NACO). Paper accepted for publication 06-Sept-2012Journal: Numer. Algebra Control Optim. 3 (2013), no. 1, 77--94Categories: math.OCKeywords: hahns symmetric quantum variational calculus, hahns symmetric variational calculus, hahn symmetric quantum calculus, necessary optimality condition, leitmanns direct methodTags: journal articleWe introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler-Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.
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arXiv:1101.3653 (Published 2011-01-19)
Higher-order Hahn's quantum variational calculus
Comments: Submitted 30-Sep-2010; revised 4-Jan-2011; accepted 19-Jan-2011; for publication in Nonlinear Analysis Series A: Theory, Methods & ApplicationsJournal: Nonlinear Analysis 75 (2012), no. 3, 1147-1157Keywords: higher-order hahns quantum variational calculus, necessary optimality condition, quantum variational problems, euler-lagrange type, hahns derivativesTags: journal articleWe prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn's derivatives of higher-order.
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arXiv:1007.0584 (Published 2010-07-04)
Euler-Lagrange equations for composition functionals in calculus of variations on time scales
Comments: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-2010Journal: Discrete Contin. Dyn. Syst. 29 (2011), no. 2, 577--593Categories: math.OCKeywords: euler-lagrange equations, composition functionals, variations, natural boundary conditions, necessary optimality conditionTags: journal articleIn this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.
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arXiv:0904.1060 (Published 2009-04-07)
Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions
Comments: 15 pagesIn this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constrain on the solution of the state equation is also considered.
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arXiv:0807.2596 (Published 2008-07-16)
Calculus of Variations on Time Scales with Nabla Derivatives
Comments: Partially presented at the Fifth World Congress of Nonlinear Analysts (WCNA-2008), Orlando, Florida, July 2-9, 2008Journal: Nonlinear Anal. 71 (2009), no. 12, e763--e773Keywords: time scales, nabla derivatives, variations, necessary optimality condition, general fundamental lemmaTags: journal articleWe prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of variations on time scales.
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Isoperimetric Problems of the Calculus of Variations on Time Scales
Comments: Changes suggested by the referee. This is the preprint version accepted (July 16, 2008) for publication in the Proceedings of the Conference on Nonlinear Analysis and Optimization, June 18-24, 2008, Technion, Haifa, Israel, and to appear in "Contemporary Mathematics"Journal: in Nonlinear Analysis and Optimization II, Contemporary Mathematics, vol. 514, Amer. Math. Soc., Providence, RI, 2010, 123--131DOI: 10.1090/conm/514Keywords: time scales, isoperimetric problems, variations, sturm-liouville eigenvalue problems, necessary optimality conditionTags: conference paper, journal articleWe prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales.
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arXiv:0801.4332 (Published 2008-01-28)
Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem
Comments: Proc. Workshop on Mathematical Control Theory and Finance, Lisbon, 10-14 April 2007, pp. 501--507Journal: Mathematical Control Theory and Finance, Springer, 2008, pp. 387--395Keywords: dead oil isotherm optimal control, discrete dead oil isotherm optimal, oil isotherm optimal control problem, necessary optimality conditionTags: journal articleWe obtain necessary optimality conditions for a semi-discretized optimal control problem for the classical system of nonlinear partial differential equations modelling the water-oil (isothermal dead-oil model).
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arXiv:0801.2123 (Published 2008-01-14)
Necessary and sufficient conditions for local Pareto optimality on time scales
Comments: 7 pagesJournal: Journal of Mathematical Sciences, Vol. 161, No. 6, 2009, 803--810Keywords: time scales, sufficient conditions, weak local pareto optimality, necessary optimality condition, multiobjective variational problemTags: journal articleWe study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales.
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arXiv:0704.0949 (Published 2007-04-06)
Conservation laws for invariant functionals containing compositions
Comments: Accepted for an oral presentation at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), to be held in Pretoria, South Africa, 22-24 August, 2007Journal: Applicable Analysis, Volume 86, Issue 9, 2007, pp. 1117-1126.Categories: math.OCKeywords: invariant functionals containing compositions, conservation laws, necessary optimality condition, generalized euler-lagrange equation, chaotic mapsTags: journal articleThe study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.