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In this paper, we present an inexact multiblock alternating direction method for the point-contact friction model of the force-optimization problem (FOP). The friction-cone constraints of the FOP are reformulated as the Cartesian product of circular cones. We focus on the convex quadratic circular-cone programming model of the FOP, which is an exact cone-programming model. Coupled with the separable convex quadratic objective function, we recast the circular-cone-programming model as a multiblock separable cone model. A parallel inexact multiblock alternating direction method is used to solve the FOP. We prove the global convergence of the proposed method. Simulation results of the three-fingered FOP are reported, which verified the efficiency of the proposed method.

Distance-weighted discrimination (DWD) is a modern margin-based classifier with an interesting geometric motivation. It was proposed as a competitor to the support vector machine (SVM). Despite many recent references on DWD, DWD is far less popular than the SVM, mainly because of computational and theoretical reasons. We greatly advance the current DWD methodology and its learning theory. We propose a novel thrifty algorithm for solving standard DWD and generalized DWD, and our algorithm can be several hundred times faster than the existing state of the art algorithm based on second-order cone programming. In addition, we exploit the new algorithm to design an efficient scheme to tune generalized DWD. Furthermore, we formulate a natural kernel DWD approach in a reproducing kernel Hubert space and then establish the Bayes risk consistency of the kernel DWD by using a universal kernel such as the Gaussian kernel. This result solves an open theoretical problem in the DWD literature. A comparison study on 16 benchmark data sets shows that data-driven generalized DWD consistently delivers higher classification accuracy with less computation time than the SVM.

The popularity of direct current (DC) networks have made their optimal power flow (OPF) problem a hot topic. With the proliferation of distributed generation, the many problems of centralized optimization methods, such as single point failure and slow response speed, have led to utilization of measures such as distributed OPF methods. The OPF problem is non-convex, which makes it difficult to obtain an optimal solution. The second-order cone programming (SOCP) relaxation method is widely utilized to make the OPF problem convex. It is difficult to guarantee its exactness, especially when line constraints are considered. This paper proposes a penalty based ADMM approach using difference-of-convex programming (DCP) to solve the non-convex OPF problem in a distributed manner. The algorithm is composed of distributed x iteration, z iteration and λ, μ iteration. Specifically, in the distributed z iteration, the active power flow injection equation of each line is formulated as a difference of two convex functions, and then the SOCP relaxation is given in a different form. If the SOCP relaxation is inexact, a penalty item is added to drive the solution to be feasible. Then, an optimal solution can be obtained using a local nonlinear programming method. Finally, simulations on a 14-bus system and the IEEE 123-bus system validate the effectiveness of the proposed approach.

Estimation of support frontiers and boundaries often involves monotone and/or concave edge data smoothing. This estimation problem arises in various unrelated contexts, such as optimal cost and production assessments in econometrics and master curve prediction in the reliability programmes of nuclear reactors. Very few constrained estimators of the support boundary of a bivariate distribution have been introduced in the literature. They are based on simple envelopment techniques which often suffer from lack of precision and smoothness. Combining the edge estimation idea of Hall, Park and Stern with the quadratic spline smoothing method of He and Shi, we develop a novel constrained fit of the boundary curve which benefits from the smoothness of spline approximation and the computational efficiency of linear programmes. Using cubic splines is also feasible and more attractive under multiple shape constraints; computing the optimal spline smoother is then formulated as a second‐order cone programming problem. Both constrained quadratic and cubic spline frontiers have a similar level of computational complexity to those of the unconstrained fits and inherit their asymptotic properties. The utility of this method is illustrated through applications to some real data sets and simulation evidence is also presented to show its superiority over the best‐known methods.

A Markov Decision Process (MDP) is a framework used for decision-making. In an MDP problem, the decision maker’s goal is to maximize the expected discounted value of future rewards while navigating through different states controlled by a Markov chain. In this paper, we focus on the case where the transition probabilities vector is deterministic, while the reward vector is uncertain and follow a partially known distribution. We employ a distributionally robust chance constraints approach to model the MDP. This approach entails the construction of potential distributions of reward vector, characterized by moments or statistical metrics. We explore two situations for these ambiguity sets: one where the reward vector has a real support and another where it is constrained to be nonnegative. In the case of a real support, we demonstrate that solving the distributionally robust chance-constrained Markov decision process is mathematically equivalent to a second-order cone programming problem for moments and ϕ-divergence ambiguity sets. For Wasserstein distance ambiguity sets, it becomes a mixed-integer second-order cone programming problem. In contrast, when dealing with nonnegative reward vector, the equivalent optimization problems are different. Moments-based ambiguity sets lead to a copositive optimization problem, while Wasserstein distance-based ambiguity sets result in a biconvex optimization problem. To illustrate the practical application of these methods, we examine a machine replacement problem and present results conducted on randomly generated instances to showcase the effectiveness of our proposed methods.

Grasping force optimization of multi-fingered robotic hands can be formulated as a convex quadratic circular cone programming problem, which consists in minimizing a convex quadratic objective function subject to the friction cone constraints and balance constraints of external force. This paper presents projection and contraction methods for grasping force optimization problems. The proposed projection and contraction methods are shown to be globally convergent to the optimal grasping force. The global convergence makes projection and contraction methods well suited to the warm-start techniques. The numerical examples show that the projection and contraction methods with warm-start version are fast and efficient.

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP), and pth order cone programming (POCP), in analogy to the algebraic degree of semidefinite programming [J.Nie,K.Ranestad,andB.Sturmfels,The algebraic degree of semidefinite programming, Math. Programm., to appear]. [PUBLICATION ABSTRACT]

The penetration of renewable generation will affect the energy utilization efficiency, economic benefit and reliability of the active distribution network (ADN). This paper proposes a time-sequence production simulation (TSPS) method for renewable generation capacity and reliability assessments in ADN considering two operational status: the normal status and the fault status. During normal operation, an optimal dispatch model is proposed to promote the renewable consumption and increase the economic benefit. When a failure occurs, the renewable generators are partitioned into islands for resilient power supply and reliability improvement. A novel dynamic island partition model is presented based on mixed integer second-order cone programming (MISOCP). The effectiveness of the proposed TSPS method is demonstrated in a standard network integrated with historical data of load and renewable generations.

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them are incomparable to the standard SDP relaxation of OPF. Extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature: (i) their solution quality is extremely close to that of the standard SDP relaxation (the best one is within 99.96% of the SDP relaxation on average for all the IEEE test cases) and consistently outperforms previously proposed convex quadratic relaxations of the OPF problem, (ii) the solutions from the strong SOCP relaxations can be directly used as a warm start in a local solver such as IPOPT to obtain a high quality feasible OPF solution, and (iii) in terms of computation times, the strong SOCP relaxations can be solved an order of magnitude faster than the standard SDP relaxation. For example, one of the proposed SOCP relaxations together with IPOPT produces a feasible solution for the largest instance in the IEEE test cases (the 3375-bus system) and also certifies that this solution is within 0.13% of global optimality, all this computed within 157.20 seconds on a modest personal computer. Overall, the proposed strong SOCP relaxations provide a practical approach to obtain feasible OPF solutions with extremely good quality within a time framework that is compatible with the real-time operation in the current industry practice.

We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) nonnegative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.