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1
Content available remote Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets
Pallaschke D., Urbański R.
Schedae Informaticae
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2012
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Vol. 21
107--125
EN
The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–Rådström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–Rådström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.
2
Content available remote Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints
El Mouatasim A., Ellaia R., Souza de Cursi E.
International Journal of Applied Mathematics and Computer Science
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2011
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Vol. 21, no 2
317-329
EN
We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.
3
Content available Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints
Wachsmuth D., Wachsmuth G.
Control and Cybernetics
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2011
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Vol. 40, no 4
1125-1158
EN
In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order.
4
Optimality and duality for nonsmooth multiobjective optimization problems with generalized V-r-invexity
Mishra S. K., Singh V., Wang S. Y., Lai K. K.
Journal of Applied Analysis
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2010
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Vol. 16, nr 1
49-58
EN
In this paper, a new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-type I functions. The generalized Karush-Kuhn-Tucker sufficient optimality conditions are proved and duality theorems are established for a non-smooth multiobjective optimization problems involving K-r-type I functions with respect to the same function η.
5
Content available remote The shooting approach in analyzing bang-bang extremals with simultaneous control switches
Felgenhauer U.
Control and Cybernetics
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2008
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Vol. 37, no 2
307-327
EN
The paper is devoted to stability investigation of optimal structure and switching points position for parametric bang-bang control problem with special focus on simultaneous switches of two control components. In contrast to problems where only simple switches occur, the switching points in general are no longer differentiable functions of input parameters. Conditions for Lipschitz stability are found which generalize known sufficient optimality conditions to nonsmooth situation. The analysis makes use of backward shooting representation of extremals, and of generalized implicit function theorems. The Lipschitz properties are illustrated for an example by constructing backward parameterized family of extremals and providing first-order switching points prediction.
6
On strict pseudoconvexity
Ivanov V., I.
Journal of Applied Analysis
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2007
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Vol. 13, nr 2
183-196
EN
The present paper provides first and second-order characterizations of a radilly lower semicontinuous strictly pseudoconvex function ∫ : X → R defined on a convex set X in the real Euclidean space Rn in twerms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived
7
Content available remote Local cone approximations in optimization
Castellani M., Pappalardo M.
Control and Cybernetics
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2007
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Vol. 36, no 3
583-600
EN
We show how to use intensively local cone approximations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theorems and error bound.
8
Content available remote Composite semi-infinite optimization
Dentcheva D., Ruszczyński A.
Control and Cybernetics
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2007
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Vol. 36, no 3
633-646
EN
We consider a semi-infinite optimization problem in Banach spaces, where both the objective functional and the constraint operator are compositions of convex nonsmooth mappings and differentiable mappings. We derive necessary optimality conditions for these problems. Finally, we apply these results to non-convex stochastic optimization problems with stochastic dominance constraints, generalizing earlier results.
9
Content available remote Random perturbation of the variable metric method for unconstrained nonsmooth nonconvex optimization
El Mouatasim A., Ellaia R., Souza de Cursi J. E.
International Journal of Applied Mathematics and Computer Science
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2006
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Vol. 16, no 4
463-474
EN
We consider the global optimization of a nonsmooth (nondifferentiable) nonconvex real function. We introduce a variable metric descent method adapted to nonsmooth situations, which is modified by the incorporation of suitable random perturbations. Convergence to a global minimum is established and a simple method for the generation of suitable perturbations is introduced. An algorithm is proposed and numerical results are presented, showing that the method is computationally effective and stable.
10
Content available remote Optimal control of constrained delay-differential inclusions with multivalued initial conditions
Mordukhovich B. S., Wang L.
Control and Cybernetics
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2003
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Vol. 32, no 3
585-609
EN
This paper studies a general optimal control problem for nonconvex delay-differential inclusions with endpoint constraints. In contrast to previous publications on this topic, we incorporate time-dependent set constraints on the initial interval, which are specific for systems with delays and provide an additional source for optimization. Our variational analysis is based on well-posed discrete approximations of constrained delay-differential inclusions by a family of time-delayed systems with discrete dynamics and perturbed constraints. Using convergence results for discrete approximations and advanced tools of nonsmooth variational analysis, we derive necessary optimality conditions for constrained delay-differential inclusions in both Euler-Lagrange and Hamiltonian forms involving nonconvex generalized differential constructions for nonsmooth functions, sets, and set-valued mappings.
11
Content available Structural optimization with a non-smooth buckling load criterion
Folgado J., Rodrigues H.
Control and Cybernetics
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1998
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Vol. 27, no 2
235-253
EN
This work addresses problems in optimal design of structures with a non-smooth buckling load criterion and in particular appliactions in layout design of plate reinforcements using a material based model and thickness beam optimization problems. Starting with the formulation of the linearized buckling problem, the optimization problem is formulated and the optimal necessary conditions are derived. Considering the possibility of non-differentiability of the objective function, the optimal necessary conditions are stated in terms of generalized gradients for non-smooth functions. The importance of the obtained result is analyzed and directional derivatives of the critical load factor obtained from the generalized gradient set definition, are compared with forward finite difference approximations. Optimization applications, to test the developments done, are presented. They are performed using a mathematical programming code, the Bundle Trust Method, which addresses the nonsmoothness of the problem.
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