On the incomplete results for the heterogeneous server problem

In this article, we show that the arguments in Rykov [9] on the optimality of a threshold routing policy when there are more than two heterogeneous servers are incomplete. AMS Subject Classifications: 93E20, 60K25, 90B22     

Maximizing throughput in finite-source parallel queue systems

Motivated by the dispatching of trucks to shovels in surface mines, we study optimal routing in a Markovian finite-source, multi-server queueing system with heterogeneous servers, each with a separate queue. We formulate the problem of routing customers to servers to maximize the system throughput as a Markov Decision Process. When the servers are homogeneous, we demonstrate that the Shortest Queue policy is optimal, and when the servers are heterogeneous, we partially characterize the optimal policy and present a near-optimal and simple-to-implement policy. We use the model to illustrate the substantial benefits of pooling, by comparing it to the permanent assignment of customers to servers.     

Optimality Conditions and Duality in Nondifferentiable Minimax Fractional Programming with Generalized Convexity

We establish sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving (F, α, ρ, d)-convexity. Subsequently, we apply the optimality conditions to formulate two types of dual problems and prove appropriate duality theorems. The authors thank the referee for valuable suggestions improving the presentation of the paper.     

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

Asymptotically optimal open-loop load balancing

In many distributed computing systems, stochastically arriving jobs need to be assigned to servers with the objective of minimizing waiting times. Many existing dispatching algorithms are basically included in the SQ(d) framework: Upon arrival of a job, \(d\ge 2\) servers are contacted uniformly at random to retrieve their state and then the job is routed to a server in the best observed state. One practical issue in this type of algorithm is that server states may not be observable, depending on the underlying architecture. In this paper, we investigate the assignment problem in the open-loop setting where no feedback information can flow dynamically from the queues back to the controller, i.e., the queues are unobservable. This is an intractable problem, and unless particular cases are considered, the structure of an optimal policy is not known. Under mild assumptions and in a heavy-traffic many-server limiting regime, our main result proves the optimality of a subset of deterministic and periodic policies within a wide set of (open-loop) policies that can be randomized or deterministic and can be dependent on the arrival process at the controller. The limiting value of the scaled stationary mean waiting time achieved by any policy in our subset provides a simple approximation for the optimal system performance.     

Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval-objective function are convex.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

First-and second-order optimality conditions for mathematical programs with vanishing constraints

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions, were recently investigated in depth. This paper enlarges the set of optimality criteria by stating first-order sufficient and second-order necessary and sufficient optimality conditions for MPVCs. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant KA1296/15-1.     

Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d?8 and d=24.     

On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.     

Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions

In this paper, new classes of generalized (F,α,ρ,d)-type I functions are introduced for differentiable multiobjective programming. Based upon these generalized functions, first, we obtain several sufficient optimality conditions for feasible solution to be an efficient or weak efficient solution. Second, we prove weak and strong duality theorems for mixed type duality.     

Synchronous working vacation policy for finite-buffer multiserver queueing system

This paper presents modeling and analysis of unreliable Markovian multiserver finite-buffer queue with discouragement and synchronous working vacation policy. According to this policy, c servers keep serving the customers until the number of idle servers reaches the threshold level d; then d idle servers take vacation altogether. Out of these d vacationing servers, d_W servers may opt for working vacation i.e. they serve the secondary customers with different rates during the vacation period. On the other hand, the remaining d − d_W = d_V servers continue to be on vacation. During the vacation of d servers, the other e = c − d servers must be present in the system even if they are idle. On returning from vacation, if the queue size does not exceed e, then these d servers take another vacation together; otherwise start serving the customers. The servers may undergo breakdown simultaneously both in regular busy period and working vacation period due to the failure of a main control unit. This main unit is then repaired by the repairman in at most two phases. We obtain the stationary performance measures such as expected queue length, average balking and reneging rate, throughput, etc. The steady state and transient behaviours of the arriving customers and the servers are examined by using matrix analytical method and numerical approach based on Runge-Kutta method of fourth order, respectively. The sensitivity analysis is facilitated for the transient model to demonstrate the validity of the analytical results and to examine the effect of different parameters on various performance indices.     

Partial Pooling in Tandem Lines with Cooperation and Blocking

For a tandem line of finite, single-server queues operating under the production blocking mechanism, we study the effects of pooling several adjacent stations and the associated servers into a single station with a single team of servers. We assume that the servers are cross-trained (so that they can work at several different stations) and that two or more servers can cooperate on the same job. For such a system, we provide sufficient conditions on the service times and sizes of the input and output buffers at the pooled station under which pooling will decrease the departure time of each job from the system (and hence increase the system throughput). We also show that pooling decreases the total number of jobs in the system at any given time and the sojourn time of each job in the system if the departure time of each job from the system is decreased by pooling and there is an arrival stream at the first station. Moreover, we provide sufficient conditions under which pooling will improve the holding cost of each job in the system incurred before any given time, and extend our results to closed tandem lines and to queueing networks with either a more general blocking mechanism or probabilistic routing. Finally, we present a numerical study aimed at quantifying the improvements in system performance obtained through pooling and at understanding which stations should be pooled to achieve the maximum benefit. Our results suggest that the improvements gained by pooling may be substantial and that the bottleneck station should be among the pooled stations in order to obtain the greatest benefit. AMS subject classification: 90B22     

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

Recently, Hachimi and Aghezzaf defined generalized (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F,α,ρ,d)-type I functions are extended to nondifferentiable functions. By utilizing the new concepts, we obtain several sufficient optimality conditions and prove mixed type and Mond-Weir type duality results for the nondifferentiable multiobjective programming problem.     

On Optimality Conditions for Generalized Semi-Infinite Programming Problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Nondifferentiable Minimax Fractional Programming Problems with (C, α, ρ, d)-Convexity

In this paper, we present necessary optimality conditions for nondifferentiable minimax fractional programming problems. A new concept of generalized convexity, called (C, α, ρ, d)-convexity, is introduced. We establish also sufficient optimality conditions for nondifferentiable minimax fractional programming problems from the viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of dual programs. This research was partially supported by NSF and Air Force grants     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

Robust penalty function method for an uncertain multi-time control optimization problems

This paper gives some new results on multi-time first-order PDE constrained control optimization problem in the face of data uncertainty (MCOPU). We obtain the robust sufficient optimality conditions for (MCOPU). Further, we construct an unconstrained multi-time control optimization problem (MCOPU)_? corresponding to (MCOPU) via absolute value penalty function method. Then, we show that the robust optimal solution to the constrained problem and a robust minimizer to the unconstrained problem are equivalent under suitable hypotheses. Moreover, we give some non-trivial examples to validate the results established in this paper.