Journal of Optimization Theory and Applications

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set X⊑E, two multifunctions Γ:X→2X and $$\Phi :X \to 2^{E^* } $$ , find $$(\hat x,\hat \phi ) \in X \times E^* $$ such that $$\hat x \in \Gamma (\hat x),\hat \phi \in \Phi (\hat x),{\text{ and }}\mathop {{\text{sup}}}\limits_{y \in \Gamma (\hat x)} \left\langle {\hat \phi ,\hat x - y} \right\rangle \leqslant 0.$$ We extend to the above problem a result established by Ricceri for the case Γ(x)≡X, where in particular the multifunction Φ is required only to satisfy the following very general assumption: each set Φ(x) is nonempty, convex, and weakly-star compact, and for each y∈X−:X the set $$\{ x \in X:\inf _{\phi \in \Phi (x)} \left\langle {\phi ,y} \right\rangle \leqslant 0\} $$ is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.

This paper presents the optimal tuning of cascade load force controllers for a parallel robot platform. A parameter search for the proposed cascade controller is difficult because there is no methodology to set the parameters and the search space is broad. The proposed parameter search scheme is based on a bat algorithm, which attracts a lot of attention in the evolutionary computation area due to the empirical evidence of its superiority in solving various nonconvex problems. The control design problem is formulated as an optimization problem under constraints. Typical constraints, such as mechanical limits on positions and maximal velocities of hydraulic actuators as well as on servo-valve positions, are included in the proposed algorithm. The simulation results indicate that the proposed optimal tuned cascade control is effective and efficient. These results clearly demonstrate that applied techniques exhibit a significant performance improvement over classical tuning methods.

This paper considers the problem of minimum-fuel interception with time constraint. The maneuver consists of using impulsive thrust to bring the interceptor from its initial orbit into a collision course with a target which is moving on a well-defined trajectory. The intercept time is either prescribed or is restricted to be less than an upper limit. The necessary conditions and the transversality conditions for optimality are discussed. The method of solution amounts to first solving a set of equations to obtain the primer vector for an initial one-impulse solution. Then, based on the information provided by the primer vector, rules are established to search for the optimal solution if the initial one-impulse trajectory is not optimal. The method is general, in the sense that it allows for solving the problem of three-dimensional interception with arbitrary motion for the target. Several numerical examples are presented, including orbital interceptions and interception at hyperbolic speeds of a ballistic missile.

In this paper, we will study optimality conditions of semi-infinite programs and generalized semi-infinite programs by employing lower order exact penalty functions and the condition that the generalized second-order directional derivative of the constraint function at the candidate point along any feasible direction for the linearized constraint set is non-positive. We consider three types of penalty functions for semi-infinite program and investigate the relationship among the exactness of these penalty functions. We employ lower order integral exact penalty functions and the second-order generalized derivative of the constraint function to establish optimality conditions for semi-infinite programs. We adopt the exact penalty function technique in terms of a classical augmented Lagrangian function for the lower-level problems of generalized semi-infinite programs to transform them into standard semi-infinite programs and then apply our results for semi-infinite programs to derive the optimality condition for generalized semi-infinite programs. We will give various examples to illustrate our results and assumptions.

The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of a proximal operator through a closed formula affects the implementability of the algorithm. In this paper, we allow in each block of the objective a further smooth convex function and propose a proximal version of the algorithm, which is achieved by equipping the algorithm with proximal terms induced by variable metrics. For suitable choices of the latter, the solving of the two subproblems in the iterative scheme can be reduced to the computation of proximal operators. We investigate the convergence of the proposed algorithm in a real Hilbert space setting and illustrate its numerical performances on two applications in image processing and machine learning.

Building on differential game theory involving asymmetric agents, an oligopoly game between two distinct groups of firms is analyzed and solved under open-loop information. One group develops Research & Development to reduce its marginal production costs and behaves fairly, whereas the other one violates intellectual property rights of the rival, using the stolen technology to reduce its own marginal costs. We investigate the effects of law enforcement in this setup, by discussing the appropriate fine to be determined and the profitability of unfair behavior. Finally, we assess how the duration of related trials can affect efficiency of enforcement policy.

Two ways of defining a well-conditioned minimization problem are introduced and related, with emphasis on the quantitative aspects. These concepts are used to study the behavior of the solution sets of minimization problems for functions with connected sublevel sets, generalizing results of Attouch-Wets in the convex case. Applications to continuity properties of subdifferentials and to projection mappings are pointed out.

An integral maximum principle is developed for a class of nonlinear systems containing time delays in state and control variables. Its proof is based on the theory of quasiconvex families of functions, originally developed by Gamkrelidze and extended by Banks. This result is used to obtain a pointwise principle of the Pontryagin type.

In this paper, we provide results concerning the optimal feedback control of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for our work. This system consists of two coupled PDEs exhibiting hyperbolic and parabolic characteristics, respectively, with the control action being modeled by a highly unbounded operator. We rigorously justify an optimal control theory for this class of problems and further characterize the optimal control through a suitable Riccati equation. This is achieved in part by exploiting recent techniques in the area of optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analysis which considers the propagation of singularities and optimal trace behavior of the solutions.