Second-order Cone Programming Problem Research Articles

A note on second order cone programming approach to two-stage network data envelopment analysis

In network data envelopment analysis (DEA) literature, the general multiplicative two-stage network model, where the objective function is the product of the two stage efficiency ratios, is highly nonlinear and is impossible to be directly transformed into linear programming problems. Chen and Zhu (2017) discovered that this model corresponds to a cone-type structure and can be transformed into a second order cone programming (SOCP) problem with a parameter. The transformed parametric SOCP problem is solved by computing a series of SOCP models when the parameter is varied. We show that Chen and Zhu's (2017) model can be directly transformed into a single SOCP problem, without the need for solving parametric SOCP problems. This greatly reduces the computational burden.

Characterizations of local upper Lipschitz property of perturbed solutions to nonlinear second-order cone programs

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping.

A computational homogenization approach for limit analysis of heterogeneous materials

The macroscopic strength domain and failure mode of heterogeneous or composite materials can be quantitatively determined by solving an auxiliary limit analysis problem formulated on a periodic representative volume element. In this paper, a novel numerical procedure based on kinematic theorem and homogenization theory for limit analysis of periodic composites is developed. The total velocity fields, instead of fluctuating (periodic) velocity, at microscopic level are approximated by finite elements, ensuring that the resulting optimization problem is similar to the limit analysis problem formulated for structures, except for additional constraints, which are periodic boundary conditions and averaging relations. The optimization problem is then formulated in the form of a standard second-order cone programming problem to be solved by highly efficient solvers. Effects of loading condition, representative volume element architecture, and fiber/air void volume fraction on the macroscopic strength of perforated and fiber reinforced composites are studied in numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.

Hopping trajectory optimization for surface exploration on small bodies

Surface exploration is an important way to improve the understanding of small bodies. Considering the irregular and weak gravity field near a small body, the movement of the surface explorer is generally achieved by hopping. In this paper, a guidance algorithm method based on convex optimization approach for pinpoint hopping movement on a small body is developed in order to improve the stability and accuracy of surface exploration. We formulate a fuel-optimal control problem for the single pinpoint hopping and convert it into a second-order cone programming (SOCP) problem which can be solved effectively by primal-dual-interior points method. A multi-hopping scenario is also proposed for the long-distance transfer. To certificate the performance of the proposed guidance algorithm, a full set of simulations are conducted and the effectiveness are analyzed.

Multi-Period Portfolio Optimization with Cone Constraints and Discrete Decisions

In this study, we consider multi-period portfolio optimization model that is formulated as a mixed-integer second-order cone programming problems (MISOCPs). The Markowitz (1952) mean/variance framework has been extended by including transaction costs, conditional value-at-risk (CVaR), diversification-by-sector and buy-in thresholds constraints. The model is obtained using a binary scenario tree that is constructed with monthly returns of the stocks from the S&P 500. We solve these models with a MATLAB based Mixed Integer Linear and Nonlinear Optimizer (MILANO). Numerical results show that we can solve small to medium-sized instances successfully, and we provide a substantial improvement in runtimes using warmstarts in outer approximation algorithm.

Information Collection Optimization in Designing Marketing Campaigns for Market Entry

Developing marketing strategies for a new product or a new target population is challenging, due to the scarcity of relevant historical data. Building on dynamic Bayesian learning, a sequential information collection optimization assists in creating new data points, within a finite number of learning phases. This procedure identifies effective advertisement design elements as well as customer segments that maximize the expected outcome of the final marketing campaign. In this paper, the marketing campaign performance is modeled by a multiplicative advertising exposure model with Poisson jumps. The intensity of the Poisson process is a function of the marketing campaign features. A forward-looking measurement policy is formulated to maximize the expected improvement in the value of information in each learning phase. Solving this information collection optimization is reduced to a mixed-integer second-order cone programming problem. A computationally efficient approach is proposed that consists in solving a sequence of mixed-integer linear optimization problems. The performance of the optimal learning policy over commonly used benchmark policies is evaluated using examples from the property and casualty insurance industry.

Robust operating zones identification for energy storage day-ahead operation

Abstract This paper presents a tool to robustly allocate the allowable operating zones of active and reactive power margins for multi-energy storage systems (ESSs) without violating typical distribution system constraints. This tool helps ESSs to manage its day-ahead energy independently without violating the power system constraints. The main contribution is considering the power uncertainties (loads and renewable energy) without taking very conservative decisions. For defining a robust operating zone (ROZ) for ESSs; first, a particle swarm optimization (PSO) algorithm detects the day-ahead worst-case power flow due to loads, renewable energy sources (RESs), and contingency uncertainties. Second, a second-order cone programming problem (SOCP) optimizer determines the maximum charge and discharge active powers for each ESS that maintains safe system operation limits (voltage limits, cables capacity, and reverse power flow limits). While an uncertainty budget designed by the fuzzy expert, the PSO uses this budget to reshape its searching space with less conservatism. Case studies with one hundred different uncertainty scenarios are conducted on a real 41-bus Canadian system. Simulation results have shown that the proposed algorithm provides robust operating zones for ESSs that consider uncertainties with reducing conservatism.

Recursive risk measures under regime switching applied to portfolio selection

In this paper, we define the conditional risk measure under regime switching and derive a class of time consistent multi-period risk measures. To do so, we describe the information process with regime switching in a product space associated with the product of two filtrations. Moreover, we show how to establish the corresponding multi-stage portfolio selection models using the time consistent multi-period risk measure for medium-term or long-term investments. Take the conditional value-at-risk measure as an example, we demonstrate the resulting multi-stage portfolio selection problem can be transformed into a second-order cone programming problem. Finally, we carry out a series of empirical tests to illustrate the superior performance of the proposed random framework and the corresponding multi-stage portfolio selection model.

LP-W- $\ell _{\infty }$ -SVD Algorithm for Direction-of-Arrival Estimation

This paper presents an effective direction-of-arrival (DOA) estimation method based on a new computation method of the $\ell _{\infty }$ -norm for a complex matrix. Making use of the new computation method of the $\ell _{\infty }$ -norm, the DOA estimation problem can be efficiently formulated as a linear programming (LP) problem. Thus, the proposed method can effectively give the DOA estimation by finding the sparse coefficients which are obtained by solving the LP problem. Except for the first singular value decomposition of the data matrix, the proposed method is effectively implemented in the light of the LP theory based on real-valued computation. It avoids solving the complicated second-order cone programming problem. Furthermore, it can primely suppress spurious peaks in DOA estimation. Simulation results demonstrate the efficiency of the presented approach.

Joint Transceiver Design for Linear MMSE Data Fusion in Coherent MAC Wireless Sensor Networks

This paper considers the design of minimal mean square error (MMSE) transceivers in a wireless sensor network. The problem is nonconvex and challenging, and previous results (with partial solutions and/or with convergence unproved) left much to be desired. Here we propose several approaches – 2 block coordinate descent (BCD), essentially cyclic multi-block method and its variants and distributive method to solve this problem. The proposed 2-BCD approach formulates the subproblem of joint beamformer optimization as a general second-order cone programming problem, which lends itself to standard numerical solvers and which requires no extra assumptions like previous works do. The proposed essentially cyclic multi-block approach further decomposes the joint beamformer design subproblem into multiple blocks, and rigorously solves each with semi-closed-form solution. The distributive algorithm optimizes transmitters in a decentralized manner and has never been considered in existing literature. The distributive algorithm has time complexity independent of number of sensors and is especially suitable for large-scale networks. All the previous BCD-based approaches left some singularity issue unattended as well as the convergence property unaddressed, our proposals are the first to provide a complete and provably-converging analytical solution. Extensive analysis and simulations demonstrate the merits of the novel approaches relative to existing alternatives.

Robust Nearfield Wideband Beamforming Design Based on Adaptive-Weighted Convex Optimization

Nearfield wideband beamformers for microphone arrays have wide applications in multichannel speech enhancement. The nearfield wideband beamformer design based on convex optimization is one of the typical representatives of robust approaches. However, in this approach, the coefficient of convex optimization is a constant, which has not used all the freedom provided by the weighting coefficient efficiently. Therefore, it is still necessary to further improve the performance. To solve this problem, we developed a robust nearfield wideband beamformer design approach based on adaptive-weighted convex optimization. The proposed approach defines an adaptive-weighted function by the adaptive array signal processing theory and adjusts its value flexibly, which has improved the beamforming performance. During each process of the adaptive updating of the weighting function, the convex optimization problem can be formulated as a SOCP (Second-Order Cone Program) problem, which could be solved efficiently using the well-established interior-point methods. This method is suitable for the case where the sound source is in the nearfield range, can work well in the presence of microphone mismatches, and is applicable to arbitrary array geometries. Several design examples are presented to verify the effectiveness of the proposed approach and the correctness of the theoretical analysis.

Optimized Full-Duplex MIMO DF Relaying With Limited Dynamic Range

This paper investigates the transceiver optimization of full-duplex multiple-input multiple-output decode-and-forward (DF) relaying by taking into account the practical factor of limited dynamic range (LDR). Focusing on the maximization of end-to-end achievable rate which involves highly coupled variables, we develop a convergent block coordinate ascent (BCA) algorithm to deal with the complicated optimization problem. In each iteration of BCA, the optimal source beamformer, relay receive beamformer, and destination receive beamformer all admit closed-form solutions. As for the optimization of relay transmit beamformer, we transform the corresponding non-convex maximin problem into a single second-order cone program problem which thus allows an efficient optimal solution. Simulations validate the remarkable performance gain attained by incorporating LDR into the full-duplex DF relaying transceiver optimization.

Fronthaul Load Balancing in Energy Harvesting Powered Cloud Radio Access Networks

Enhanced with wireless power transfer capability, cloud radio access network (C-RAN) enables energy-restrained mobile devices to function uninterruptedly. Beamforming of C-RAN has potential to improve the efficiency of wireless power transfer, in addition to transmission data rates. In this paper, we design the beamforming jointly for data transmission and energy transfer, under finite fronthaul capacity of C-RAN. A non-convex problem is formulated to balance the fronthaul requirements of different remote radio heads (RRHs). Norm approximations and relaxations are carried out to convexify the problem to second-order cone programming (SOCP). To improve the scalability of the design to large networks, we further decentralize the SOCP problem using the alternating direction multiplier method (ADMM). A series of reformulations and transformations are conducted, such that the resultant problem conforms to the state-of-the-art ADMM solver and can be efficiently solved in real time. Simulation results show that the distributed algorithm can remarkably reduce the time complexity without compromising the fronthaul load balancing of its centralized counterpart. The proposed algorithms can also reduce the fronthaul bandwidth requirements by 25% to 50%, compared with the prior art.

An alternative approach for a distance inequality associated with the second-order cone and the circular cone

It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.

Outage Constrained Robust Beamforming for Secure Broadcasting Systems With Energy Harvesting

In this paper, we investigate simultaneous wireless information and power transfer systems for multiuser multiple-input single-output secure broadcasting channels. Considering imperfect channel state information, we introduce a robust secure beamforming design, where the transmit power is minimized subject to the secrecy rate outage probability constraint for legitimate users and the harvested energy outage probability constraint for energy harvesting receivers. The original problem is non-convex due to the presence of the probabilistic constraints. With the aid of Bernstein-type inequalities, we transform the outage constraints into the deterministic forms. Based on a successive convex approximation (SCA) method, we propose a low-complexity approach, which reformulates the original problem as a second-order cone programming problem. Also, we prove the convergence of the SCA-based iterative algorithm. Simulation shows that the proposed scheme outperforms the conventional method with lower complexity.

Geometry of Power Flows and Convex-relaxed Power Flows in Distribution Networks with High Penetration of Renewables

Renewable energies are increasingly integrated in electric distribution networks and will cause severe overvoltage issues. Smart grid technologies make it possible to use coordinated control to mitigate the overvoltage issues and the optimal power flow (OPF) method is proven to be efficient in the applications such as curtailment management and reactive power control. Nonconvex nature of the OPF makes it difficult to solve and convex relaxation is a promising method to solve the OPF very efficiently. This paper investigates the geometry of the power flows and the convex-relaxed power flows when high penetration level of renewables is present in the distribution networks. The geometry study helps understand the fundamental nature of the OPF and its convex-relaxed problem, such as the second-order cone programming (SOCP) problem. A case study based on a three-node system is used to illustrate the geometry profile of the feasible sub-injection (injection of nodes excluding the root/substation node) region.

Cutting planes based relaxed optimal power flow in active distribution systems

Optimal power flow (OPF) has played a significant role in the design, planning and operation of active distribution systems (ADSs). Due to its ability to achieve the optimality with higher computational efficiency, the second order cone programming (SOCP) based on branch flow model (BFM) has received an increasing attention in recent years. However, various sufficient conditions and assumptions are required to ensure the relaxation exactness in the existing literatures. In this study, we introduce the cutting planes to tackle this exactness challenge for general distribution networks especially the ADS with high renewable penetration. Firstly, a typical operation optimization model is presented as an example of OPF in ADS. A general branch flow model based relaxed optimal power flow (BFM-ROPF) model is then formulated as a SOCP problem after conic relaxation. According to these conditions with the objectives which are not monotonously increasing in power injections or branch currents, a total power loss cut (TPLC) is introduced to ensure the conic relaxation exactness. Moreover, a leaf branch current cut (LBCC) is incorporated to prevent the inexactness of SOCR in some leaf branches. Afterwards, the proof of the cutting planes is given to guarantee the optimality and the relaxation exactness of the BFM-ROPF model even though power loss is not included in the objectives. Numerical results based on 33-bus and 69-bus systems are given to verify the effectiveness of the proposed method.

Customized Real-Time Interior-Point Methods for Onboard Powered-Descent Guidance

This paper presents a new onboard-implementable, real-time convex optimization-based powered-descent guidance algorithm for planetary pinpoint landing. Earlier work provided the theoretical basis of convexification, the equivalent representation of the fuel-optimal pinpoint landing trajectory optimization problem with nonconvex control constraints as a convex optimization problem. Once the trajectory optimization problem is convexified, interior-point method algorithms can be used to solve the problem to global optimality. Though having this guarantee of convergence motivated earlier convexification results, there were no real-time interior point method algorithms available for the computation of optimal trajectories on flight computers. This paper presents the first such algorithm developed for onboard use and flight-tested on a terrestrial rocket with the NASA Jet Propulsion Laboratory and the NASA Flight Opportunities Program in 2013. First, earlier convexification results are summarized and the resulting second-order cone-programming problem for fuel-optimal trajectory optimization is presented. Then, the proposed, fairly generic, second-order cone-programming interior point method algorithm is presented in detail with an overview of the customization process for real-time computations. Customization exploits a specific problem structure to increase the computational speed, which is shown to decrease run times by two to three orders of magnitude in many applications. A new convexification result for maximal-divert trajectories with active velocity constraints is also presented herein.

Second-Order Cone Programming Based Robust Blind Multiuser Detector for CDMA Systems

In this paper, we propose a robust blind multiuser detector for code-division multiple access (CDMA) systems against signature waveform mismatch (SWM) derived from the influences of time asynchronization or channel distortion. This blind detection method with SWM problem is formulated as blind source separation (BSS) model subject to the second-order cone (SOC) constraint. The resulting blind separation based on SOC programming problem is solved by approximate negentropy maximization using quasiNewton iterative methods. Theoretical analysis and simulation results show that the performance of the proposed blind detector is superior to those of the existing methods.

A projected-based neural network method for second-order cone programming

A projected-based neural network method for second-order cone programming is proposed. The second-order cone programming is transformed into an equivalent projection equation. The projection on the second-order cone is simple and costs less computation time. We prove that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the second-order cone programming problem. The simulation experiments show our method is an efficient method for second-order cone programming problems.