Second-order Cone Optimization Problem Research Articles

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

An Irregular Tiled Array Technique for Microwave Wireless Power Transmission

An irregular tiled array technique with the maximization of beam collection efficiency (BCE) for microwave wireless power transmission (WPT) is addressed in this paper, where the radiating elements are irregularly partitioned into multiple subarrays for reducing the number of radio frequency (RF) chain. The system architectures of WPT based on irregular tiled arrays are analyzed for the exhibition of its low-cost advantages. To achieve a good trade-off between the manufacturing cost and beampattern, a mixed-integer optimization problem (MIOP) accounting for the maximization of BCE satisfying the subarray tiling configuration, scanning direction and peak sidelobe level (SLL) constraints is established. By equivalently reformulating the beampattern, the MIOP in terms of subarray tiling configuration and subarray excitations is decoupled, and a two-stage solution method can be implemented. Firstly, resort to ‘generalized BCE’, the binary quadratic optimization problem (BQOP) for maximizing the BCE at considered scanning angles while guaranteeing the full aperture coverage is reduced to a binary second-order cone optimization problem. After the subarray tiling configuration is determined, the non-convex Rayleigh entropy maximization problem at a given scanning angle with respect to subarray complex excitations while satisfying peak SLL constraint is globally solved by using iterative convex approximation algorithm. Owing to the high efficiency of convex optimization, the BCE is able to monotonically increase to a stable value. Several numerical results with different array size and subarray shapes are provided to assess the effectiveness of the proposed method by comparing to the competitive counterparts in the open literatures.

Robust Control Allocation for Space Inertial Sensor under Test Mass Release Phase with Overcritical Conditions

This paper proposes a robust control allocation for the capture control of the space inertial sensor's test mass under overcritical conditions. Uncertainty factors of the test mass control system under the overcritical condition are analyzed first, and a 6-DOF test mass dynamics model with system uncertainty is established. Subsequently, a time-varying weight function is designed to coordinate the allocation of 6-DOF generalized forces. Moreover, a robust control allocation method is proposed to distribute the commanded forces and torques into individual electrodes in an optimal manner, which takes into account the system uncertainties. This method transforms the robust control allocation problem into a second-order cone optimization problem, and its dual problem is introduced to simplify the computational complexity and improve the solving efficiency. Numerical simulation results are presented to illustrate and highlight the fine performance benefits obtained using the proposed robust control allocation method, which improves capture efficiency, increases the security margin and reduces allocation errors.

COMPLEMENTARY ENERGY PRINCIPLE FOR CABLE NETWORKS IN TERMS OF FORCE VECTOR VARIABLE

An alternative approach to the variation formulation of static equilibrium problem for cable networks is proposed. The dual minimization principle is derived in terms of force variables. The full vector force in the cable member is chosen as the dual variable. Unlike the previously proposed treatments of the problem using the scalar axial force variable the elastic energy retains the most general form until the closing steps. It is only assumed to be a convex function of the member end-to-end vector. As a result the nonsmooth asymmetric response of the cable does not appear in the form of inequality in the very beginning. Instead it shows up in the final form of the complementary energy functional, which turn out to be non-differentiable for zero member forces. The kinematic relations between member elongations and nodal displacements convert into the force equilibrium conditions in terms of nodal reactions and member force vectors. The Dirichlet boundary conditions appear in the dual form as a linear term in the complementary energy functional. For the linear elastic cables this approach lead to a second-order cone optimization problem with two sets conical constraints. Such formulation provides flexibility and thus can be potentially extended to account for phenomena such as inter-fiber friction and adhesion. Keywords: complementary energy, variational principles, cable networks

Second-order cone and semidefinite methods for the bisymmetric matrix approximation problem

Approximating the closest positive semi-definite bisymmetric matrix using the Frobenius norm to a data matrix is important in many engineering applications, communication theory and quantum physics. In this paper, we will use the interior point method to solve this problem. The problem will be reformulated into various forms, in the beginning as a semi-definite programming problem and later, into the form of a mixed semidefintie and second-order cone optimization problem. Numerical results comparing the efficiency of these methods with the alternating projection algorithm will be reported.

Influence of electric vehicle distributed energy storage access on safety and stability of distribution network

This paper proposes a distributed energy storage control strategy for electric vehicles to improve the security and stability of distribution network when electric vehicles are connected. Firstly, the load characteristics of electric vehicles are investigated, and the optimal power flow model including energy storage power station, electric vehicle charging station considering V2G and distribution network is established. The objective function is established to minimize network loss, voltage deviation and load peak to peak. The problem is transformed into a mixed integer second-order cone optimization problem for solution, and based on the analysis of distributed energy storage model and constraints, the distributed energy storage control strategy of electric vehicle is established. Through the solution analysis in IEEE33 node system, it can be concluded that using this strategy can improve the power quality of distribution network and effectively improve the security and stability of distribution network.

Constrained Dual-Level Bandit for Personalized Impression Regulation in Online Ranking Systems

Impression regulation plays an important role in various online ranking systems, e.g. , e-commerce ranking systems always need to achieve local commercial demands on some pre-labeled target items like fresh item cultivation and fraudulent item counteracting while maximizing its global revenue. However, local impression regulation may cause “butterfly effects” on the global scale, e.g. , in e-commerce, the price preference fluctuation in initial conditions (overpriced or underpriced items) may create a significantly different outcome, thus affecting shopping experience and bringing economic losses to platforms. To prevent “butterfly effects”, some researchers define their regulation objectives with global constraints, by using contextual bandit at the page-level that requires all items on one page sharing the same regulation action, which fails to conduct impression regulation on individual items. To address this problem, in this article, we propose a personalized impression regulation method that can directly makes regulation decisions for each user-item pair. Specifically, we model the regulation problem as a C onstrained D ual-level B andit (CDB) problem, where the local regulation action and reward signals are at the item-level while the global effect constraint on the platform impression can be calculated at the page-level only. To handle the asynchronous signals, we first expand the page-level constraint to the item-level and then derive the policy updating as a second-order cone optimization problem. Our CDB approaches the optimal policy by iteratively solving the optimization problem. Experiments are performed on both offline and online datasets, and the results, theoretically and empirically, demonstrate CDB outperforms state-of-the-art algorithms.

The rate of convergence of proximal method of multipliers for second-order cone optimization problems

In this paper we consider a proximal method of multipliers (PMM) for a nonlinear second-order cone optimization problem. With the assumptions of constraint nondegeneracy, strict complementarity and second-order sufficient condition, we estimate the local convergence rate of PMM to be linear or superlinear, which depends on the strategy of parameter selection.

A new approximation hierarchy for polynomial conic optimization

In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Polya’s Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615–625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.

Optimal Infrastructure Planning of Active Distribution Networks Complying With Service Restoration Requirements

For a fault occurring in a feeder of an active distribution system, the targeted restoration strategy consists of activating, in a first stage, clusters of nodes supplied each by a unique resource. This resource could be a DG or a tie-switch (tie-line) that provides power from a neighboring feeder. To reinforce the security of supply, in a second stage, each DG cluster is connected to a tie-switch cluster to form a so-called large cluster. Therefore, the problem presented in this paper aims at determining the optimal numbering, sizing, and location of potential distributed energy resources as well as the optimal border of the associated clusters. It is formulated in term of a convex second-order cone optimization problem where the objective is to find the best trade-off between picking up as many loads as possible whatever the fault location and the investment plus the maintenance costs. This problem has been solved in the case of a modified IEEE 34-bus system considering different potential infrastructure scenarios.

A relaxed-certificate facial reduction algorithm based on subspace intersection

A “facial reduction”-like regularization algorithm is established for general conic optimization problems by relaxing requirements on the reduction certificates. This yields a rapid subspace reduction algorithm challenged only by representational issues of the regularized cone. A condition for practical usage is analyzed and shown to always be satisfied for single second-order cone optimization problems. Should the condition fail on some other class of instances, only partial regularization is achieved based on the success of the individual subspace intersection.

Efficient formulations of the material identification problem using full-field measurements

The material identification problem addressed consists of determining the constitutive parameters distribution of a linear elastic solid using displacement measurements. This problem has been considered in important applications such as the design of methodologies for breast cancer diagnosis. Since the resolution of real life problems involves high computational costs, there is great interest in the development of efficient methods. In this paper two new efficient formulations of the problem are presented. The first formulation leads to a second-order cone optimization problem, and the second one leads to a quadratic optimization problem, both allowing the resolution of the problem with high efficiency and precision. Numerical examples are solved using synthetic input data with error. A regularization technique is applied using the Morozov criterion along with an automatic selection strategy of the regularization parameter. The proposed formulations present great advantages in terms of efficiency, when compared to other formulations that require the application of general nonlinear optimization algorithms.

Robust two-stage stochastic linear programs with moment constraints

We consider the two-stage stochastic linear programming model, in which the recourse function is a worst case expected value over a set of probabilistic distributions. These distributions share the same first- and second-order moments. By using duality of semi-infinite programming and assuming knowledge on extreme points of the dual polyhedron of the constraints, we show that a deterministic equivalence of the two-stage problem is a second-order cone optimization problem. Numerical examples are presented to show non-conservativeness and computational advantage of this approach.

On the Identification of the Optimal Partition of Second Order Cone Optimization Problems

This paper discusses the identification of the optimal partition of second order cone optimization (SOCO). By giving some condition numbers which only depend on the SOCO problem itself, we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition $\mathcal{B}, \mathcal{N}, \mathcal{R}$, and $\mathcal{T}$ for SOCO problems can be identified along the central path when the barrier parameter $\mu$ is small enough. Then we generalize the results to a specific neighborhood of the central path.

Two-stage stochastic linear programs with incomplete information on uncertainty

Two-stage stochastic linear programming is a classical model in operations research. The usual approach to this model requires detailed information on distribution of the random variables involved. In this paper, we only assume the availability of the first and second moments information of the random variables. By using duality of semi-infinite programming and adopting a linear decision rule, we show that a deterministic equivalence of the two-stage problem can be reformulated as a second-order cone optimization problem. Preliminary numerical experiments are presented to demonstrate the computational advantage of this approach.

A VaR Black–Litterman model for the construction of absolute return fund-of-funds

The objective is to construct fund-of-funds (FoFs) that follow an absolute return strategy and meet the requirements imposed by the value-at-risk (VaR) market risk measure. We propose the VaR Black–Litterman model which accounts for the VaR and trading (diversification, buy-in threshold, liquidity, currency) requirements. The model takes the form of a probabilistic integer, non-convex optimization problem. We first derive a deterministic reformulation of the probabilistic problem, which, depending on the information on the probability distribution of the FoF return, is the equivalent, or a close approximation, of the original problem. We then show that the continuous relaxation of the reformulated problem is a second-order cone optimization problem for a wide range of probability distributions. Finally, we use a specialized nonlinear branch-and-bound algorithm which implements the portfolio risk branching rule to construct the optimal FoF. The practical relevance of the model and solution method is shown by their use by a financial institution for the construction of several FoFs that are now traded worldwide. The computational study attests that the proposed algorithmic technique is very efficient, outperforming, in terms of both speed and robustness, three state-of-the-art alternative solution methods and solvers.

Robust Approximation to Multiperiod Inventory Management

We propose a robust optimization approach to address a multiperiod inventory control problem under ambiguous demands, that is, only limited information of the demand distributions such as mean, support, and some measures of deviations. Our framework extends to correlated demands and is developed around a factor-based model, which has the ability to incorporate business factors as well as time-series forecast effects of trend, seasonality, and cyclic variations. We can obtain the parameters of the replenishment policies by solving a tractable deterministic optimization problem in the form of a second-order cone optimization problem (SOCP), with solution time; unlike dynamic programming approaches, it is polynomial and independent on parameters such as replenishment lead time, demand variability, and correlations. The proposed truncated linear replenishment policy (TLRP), which is piecewise linear with respect to demand history, improves upon static and linear policies, and achieves objective values that are reasonably close to optimal.

Goal-Driven Optimization

We develop a goal-driven stochastic optimization model that considers a random objective function in achieving an aspiration level, target, or goal. Our model maximizes the shortfall-aware aspiration-level criterion, which encompasses the probability of success in achieving the aspiration level and an expected level of underperformance or shortfall. The key advantage of the proposed model is its tractability. We can obtain its solution by solving a small collection of stochastic linear optimization problems with objectives evaluated under the popular conditional-value-at-risk (CVaR) measure. Using techniques in robust optimization, we propose a decision-rule-based deterministic approximation of the goal-driven optimization problem by solving subproblems whose number is a polynomial with respect to the accuracy, with each subproblem being a second-order cone optimization problem (SOCP). We compare the numerical performance of the deterministic approximation with sampling-based approximation and report the computational insights on a multiproduct newsvendor problem.

Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones

The paper analyzes the rate of local convergence of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under the constraint nondegeneracy condition and the strong second order sufficient condition, we demonstrate that the sequence of iterate points generated by the augmented Lagrangian method locally converges to a local minimizer at a linear rate, whose ratio constant is proportional to 1/τ with penalty parameter τ not less than a threshold \(\hat{\tau}>0\) . Importantly and interestingly enough, the analysis does not require the strict complementarity condition.

Random Volumetric MRI Trajectories via Genetic Algorithms

A pseudorandom, velocity-insensitive, volumetric k-space sampling trajectory is designed for use with balanced steady-state magnetic resonance imaging. Individual arcs are designed independently and do not fit together in the way that multishot spiral, radial or echo-planar trajectories do. Previously, it was shown that second-order cone optimization problems can be defined for each arc independent of the others, that nulling of zeroth and higher moments can be encoded as constraints, and that individual arcs can be optimized in seconds. For use in steady-state imaging, sampling duty cycles are predicted to exceed 95 percent. Using such pseudorandom trajectories, aliasing caused by under-sampling manifests itself as incoherent noise. In this paper, a genetic algorithm (GA) is formulated and numerically evaluated. A large set of arcs is designed using previous methods, and the GA choses particular fit subsets of a given size, corresponding to a desired acquisition time. Numerical simulations of 1 second acquisitions show good detail and acceptable noise for large-volume imaging with 32 coils.