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Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new class of interest to the optimization community and practitioners, in which certain variables are required to be integers. In this paper, we describe five applications that lead to stochastic mixed-integer second-order cone programming problems. Additionally, we present solution algorithms for solving stochastic mixed-integer second-order cone programming using cuts and relaxations by combining existing algorithms for stochastic second-order cone programming with extensions of mixed-integer second-order cone programming. The applications, which are the focus of this paper, include facility location, portfolio optimization, uncapacitated inventory, battery swapping stations, and berth allocation planning. Considering the fact that mixed-integer programs are usually known to be NP-hard, bringing applications to the surface can detect tractable special cases and inspire for further algorithmic improvements in the future.

Several logarithmic-barrier interiors-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the ge...

In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature.

We present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton’s method is used to compute the descent direction. The main contribution of this paper is that it uniquely uses the so-called majorant functions as an efficient alternative to line search methods to determine the displacement step along the direction while solving second-order cone programs. The efficiency of our method is shown by presenting numerical experiments.

Alzalg (J Optim Theory Appl 163(1):148–164, 2014) derived a homogeneous self-dual algorithm for stochastic second-order cone programs with finite event space. In this paper, we derive an infeasible interior-point algorithm for the same stochastic optimization problem by utilizing the work of Rangarajan (SIAM J Optim 16(4), 1211–1229, 2006) for deterministic symmetric cone programs. We show that the infeasible interior-point algorithm developed in this paper has complexity less than that of the homogeneous self-dual algorithm mentioned above. We implement the proposed algorithm to show that they are efficient.

In this paper, we develop a primal-dual central trajectory interior-point algorithm for symmetric programming problems and establish its complexity analysis. The main contribution of the paper is that it uniquely equips the central trajectory algorithm with various selections of the displacement step while solving symmetric programming. To show the efficiency of the proposed algorithm, these selections of calculating the displacement step are compared in numerical examples for second-order cone programming, which is a special case of symmetric programming.

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization pro...

Alzalg, Baha/0000-0002-1839-8083; Momani, Shaher M./0000-0002-6326-8456; WOS: 000413785300005 In a recent paper (Zeb et al. in Appl Math Model 37(7):5326-5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders’ decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao’s results [G. Zhao, A log barrier method with Benders’ decomposition for solving two-stage stochastic linear programs, Math. Program. Ser. A 90 (2001) 507–536] for two-stage stochastic linear programs, and Mehrotra and Ozevin’s results [S. Mehrotra, M.G. Ozevin, Decomposition-based interior point methods for two-stage stochastic semidefinite programming, SIAM J. Optim. 18 (1) (2007) 206–222] for two-stage stochastic semidefinite programs.

Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space have been proposed by Jin et al. in [17]. Alzalg [8] has adopted their work to derive homogeneous selfdual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these to derive homogeneous selfdual algorithms for stochastic programs with finite event space over the much wider class of all symmetric cones. They include among others, stochastic semidefinite programs and stochastic second-order cone programs.

In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider the generalized Euler method (GEM) and multi-step generalized differential transform method (MSGDTM). We also show the unique positive solution for this model and present numerical results graphically.

Alzalg, Baha/0000-0002-1839-8083; Khan, Madad/0000-0003-0044-961X WOS: 000374565100001 PubMed: 27105426 In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider the generalized Euler method (GEM) and multi-step generalized differential transform method (MSGDTM). We also show the unique positive solution for this model and present numerical results graphically.

The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize a cost function and satisfy network constraints. We analyze the optimal power flow problem subject to contingency constraints and investigate the relationship between the cost of the optimal power flow problem and network topology. We find that when the network topology is that of a small world graph or a scale-free graph, the optimal power flow problem is robust in terms of satisfying contingency constraints.

In this paper we propose multi-order cone programs (MOCPs) as a new class of convex nonlinear optimization problems that includes linear programs, (convex) quadratic programs second-order cone programs and, more generally, pth-order cone programs as special cases. In MOCPs we minimize a linear objective function over the intersection of an affine set and a product of multi-order cones. We refer to them as deterministic multi-order cone programs (DMCOPs) since data defining them are deterministic. We present the definition of DMOCPs in primal and dual standard forms. Then we introduce two-stage stochastic multi-order cone programs (SMOCPs) (with recourse) to handle uncertainty in data defining DMOCPs and deterministic mixed integer multi-order cone programs (DMIMOCPs) to handle DMOCPs with integer-valued variables. We describe an applicational setting and present DMOCP, SMOCP and DMIMOCP models arising in that setting.

Jin et al. (in J. Optim. Theory Appl. 155:1073---1083, 2012) proposed homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space. In this paper, we utilize their work to derive homogeneous self-dual algorithms for stochastic second-order cone programs with finite event space. We also show how the structure in the stochastic second-order cone programming problems may be exploited so that the algorithms developed for these problems have less complexity than the algorithms developed for stochastic semidefinite programs mentioned above.