49,213 research outputs found
Second-Order Karush-Kuhn-Tucker Optimality Conditions for Vector Problems with Continuously Differentiable Data and Second-Order Constraint Qualifications
Some necessary and sufficient optimality conditions for inequality
constrained problems with continuously differentiable data were obtained in the
papers [I. Ginchev and V.I. Ivanov, Second-order optimality conditions for
problems with C\sp{1} data, J. Math. Anal. Appl., v. 340, 2008, pp.
646--657], [V.I. Ivanov, Optimality conditions for an isolated minimum of order
two in C\sp{1} constrained optimization, J. Math. Anal. Appl., v. 356, 2009,
pp. 30--41] and [V. I. Ivanov, Second- and first-order optimality conditions in
vector optimization, Internat. J. Inform. Technol. Decis. Making, 2014, DOI:
10.1142/S0219622014500540].
In the present paper, we continue these investigations. We obtain some
necessary optimality conditions of Karush--Kuhn--Tucker type for scalar and
vector problems. A new second-order constraint qualification of Zangwill type
is introduced. It is applied in the optimality conditions.Comment: 1
Pareto optimality conditions and duality for vector quadratic fractional optimization problems
One of the most important optimality conditions to aid in solving a vector optimization problem is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions on the objective functions and on the constraint set. The present work is concerned with the constrained vector quadratic fractional optimization problem. It shows that sufficient Pareto optimality conditions and the main duality theorems can be established without the assumption of generalized convexity in the objective functions, by considering some assumptions on a linear combination of Hessian matrices instead. The main aspect of this contribution is the development of Pareto optimality conditions based on a similar second-order sufficient condition for problems with convex constraints, without convexity assumptions on the objective functions. These conditions might be useful to determine termination criteria in the development of algorithms.Coordenação de aperfeiçoamento de pessoal de nivel superior (Brasil)Ministerio de Ciencia y TecnologÃaConselho Nacional de Desenvolvimento CientÃfico e Tecnológico (Brasil)Fundação de Amparo à Pesquisa do Estado de São Paul
Primal-Dual Stability in Local Optimality
Much is known about when a locally optimal solution depends in a
single-valued Lipschitz continuous way on the problem's parameters, including
tilt perturbations. Much less is known, however, about when that solution and a
uniquely determined multiplier vector associated with it exhibit that
dependence as a primal-dual pair. In classical nonlinear programming, such
advantageous behavior is tied to the combination of the standard strong
second-order sufficient condition (SSOC) for local optimality and the linear
independent gradient condition (LIGC) on the active constraint gradients. But
although second-order sufficient conditions have successfully been extended far
beyond nonlinear programming, insights into what should replace constraint
gradient independence as the extended dual counterpart have been lacking.
The exact answer is provided here for a wide range of optimization problems
in finite dimensions. Behind it are advances in how coderivatives and strict
graphical derivatives can be deployed. New results about strong metric
regularity in solving variational inequalities and generalized equations are
obtained from that as well
Optimality conditions applied to free-time multi-burn optimal orbital transfers
While the Pontryagin Maximum Principle can be used to calculate candidate
extremals for optimal orbital transfer problems, these candidates cannot be
guaranteed to be at least locally optimal unless sufficient optimality
conditions are satisfied. In this paper, through constructing a parameterized
family of extremals around a reference extremal, some second-order necessary
and sufficient conditions for the strong-local optimality of the free-time
multi-burn fuel-optimal transfer are established under certain regularity
assumptions. Moreover, the numerical procedure for computing these optimality
conditions is presented. Finally, two medium-thrust fuel-optimal trajectories
with different number of burn arcs for a typical orbital transfer problem are
computed and the local optimality of the two computed trajectories are tested
thanks to the second-order optimality conditions established in this paper
On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints
The problem of minimizing convex functionals of probability distributions is
solved under the assumption that the density of every distribution is bounded
from above and below. A system of sufficient and necessary first-order
optimality conditions as well as a bound on the optimality gap of feasible
candidate solutions are derived. Based on these results, two numerical
algorithms are proposed that iteratively solve the system of optimality
conditions on a grid of discrete points. Both algorithms use a block coordinate
descent strategy and terminate once the optimality gap falls below the desired
tolerance. While the first algorithm is conceptually simpler and more
efficient, it is not guaranteed to converge for objective functions that are
not strictly convex. This shortcoming is overcome in the second algorithm,
which uses an additional outer proximal iteration, and, which is proven to
converge under mild assumptions. Two examples are given to demonstrate the
theoretical usefulness of the optimality conditions as well as the high
efficiency and accuracy of the proposed numerical algorithms.Comment: 13 pages, 5 figures, 2 tables, published in the IEEE Transactions on
Signal Processing. In previous versions, the example in Section VI.B
contained some mistakes and inaccuracies, which have been fixed in this
versio
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