Cone Constraints Research Articles

A Decentralized Primal-Dual Method for Constrained Minimization of a Strongly Convex Function

We propose decentralized primal-dual methods for cooperative multiagent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific convex functions over conic constraint sets defined by agent-specific nonlinear functions; hence, the optimal consensus decision should lie in the intersection of these private sets. Under the strong convexity assumption, we provide convergence rates for suboptimality, infeasibility, and consensus violation in terms of the number of communications required; examine the effect of underlying network topology on the convergence rates.

Chance-Constrained Regulation Capacity Offering for HVAC Systems Under Non-Gaussian Uncertainties With Mixture-Model-Based Convexification

Heating, ventilation, and air-conditioning (HVAC) systems are ideal demand-side flexible resources to provide regulation services. However, finding the best hourly regulation capacity offers for HVAC systems in a power market ahead of time is challenging because they are affected by non-Gaussian uncertainties from regulation signals. Moreover, since HVAC systems need to frequently regulate their power according to regulation signals, numerous thermodynamic constraints are introduced, leading to a huge computational burden. This paper proposes a tractable chance-constrained model to address these challenges. It first develops a temporal compression approach, in which the extreme indoor temperatures in the operating hour are estimated and restricted in the comfortable range so that the numerous thermodynamic constraints can be compressed into only a few ones. Then, a novel convexification method is proposed to handle the non-Gaussian uncertainties. This method leverages the Gaussian mixture model to reformulate the chance constraints with non-Gaussian uncertainties on the left-hand side into deterministic non-convex forms. We further prove that these non-convex forms can be approximated by mixed-integer second-order cone constraints that can be efficiently solved by off-the-shelf solvers. The optimality gap because of this approximation is marginal under mild conditions. Numerical experiments are conducted to validate the superiority of the proposed method.

Strong Formulations for Distributionally Robust Chance-Constrained Programs with Left-Hand Side Uncertainty Under Wasserstein Ambiguity

Distributionally robust chance-constrained programs (DR-CCPs) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-M–based mixed-integer programming reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as problem size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the problem size. Our focus is on ambiguity sets under the so-called left-hand side uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-M formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-M coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning. Funding: This work was supported by the DARPA Lagrange Program [Grant N660011824020], the National Science Foundation [Grant 1740707], the Office of Naval Research [Grant N00014-19-1-2321], and the Institute for Basic Science [Grants IBS-R029-C1 and IBS-R029-Y2].

Short Communication: Cone-Constrained Monotone Mean-Variance Portfolio Selection under Diffusion Models

We consider monotone mean-variance (MMV) portfolio selection problems with a conic convex constraint under diffusion models, and their counterpart problems under mean-variance (MV) preferences. We obtain the precommitted optimal strategies to both problems in closed form and find that they coincide, with and without the presence of the conic constraint. This result generalizes the equivalence between MMV and MV preferences from nonconstrained cases to a specific constrained case. A comparison analysis reveals that the orthogonality property under the conic convex set is a key to ensuring the equivalence result.

Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment.

Given a convex and differentiable objective Q(M) for a real symmetric matrix M in the positive definite (PD) cone-used to compute Mahalanobis distances-we propose a fast general metric learning framework that is entirely projection-free. We first assume that M resides in a space S of generalized graph Laplacian matrices corresponding to balanced signed graphs. M ∈ S that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, S includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given M ∈ S and diagonal matrix S, where Sii = 1/vi and v is the first eigenvector of M, we prove that Gershgorin disc left-ends of similarity transform B = SMS-1 are perfectly aligned at the smallest eigenvalue λmin. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in M can be solved efficiently as linear programs via the Frank-Wolfe method. We update v using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in M are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.

Fitting second-order cone constraints to microbial growth data

Second-order cone programming is a highly tractable convex optimization class. In this paper, we fit general second-order cone constraints to data. This is of use when one must solve large-scale, nonlinear optimization problems, but modeling is either impractical or does not lead to second-order cone or otherwise tractable constraints. Our motivating application is biochemical process optimization, in which we seek to fit second-order cone constraints to microbial growth data. The fitting problem is nonconvex. We solve it using the concave–convex procedure, which takes the form of a sequence of second-order cone programs. We validate our approach on simulated and experimental microbial growth data, and compare its performance with conventional nonlinear least-squares fitting.

New Notions of Proper Efficiency in Set Optimization with the Set Criterion

In this paper, we introduce new notions of proper efficiency in the sense of Henig for a set optimization problem by using the set criterion of solution. The relationships between them are studied. Also, we compare these concepts with the homologous ones given by considering the vector criterion. Finally, a Lagrange multiplier rule for Henig proper solutions of a set optimization problem with a cone constraint is obtained under convexity hypotheses. Illustrative examples are also given.

Convex Approach to Covariance Control with Application to Stochastic Low-Thrust Trajectory Optimization

This paper outlines a novel approach to the design of optimal space trajectories under significant uncertainty. Finite-horizon covariance control, i.e., the steering of a system from an initial probability distribution to a desired one at a prescribed time, is employed to plan an optimal nominal path along with a robust feedback controller that compensates for exogenous in-flight disturbances. A mindful convexification strategy is devised to recast the nonlinear covariance control problem as a deterministic convex optimization problem. The convexification is based on a convenient change of variables that allows for relaxing the covariance matrix discrete-time propagation into a set of semidefinite cone constraints. While featuring a larger feasible space, the relaxed problem shares the same optimal solution as the original one. The lossless property of the relaxation is analytically proven by leveraging optimal control theory and discussed on the basis of numerical experiments. Monte Carlo campaigns are carried out to validate the in-flight performance of the attained control policies.

Bounds-constrained polynomial approximation using the Bernstein basis

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of constructioning such approximations using polynomials in the Bernstein basis. We consider a family of inequality-constrained quadratic programs. In the univariate case, a quadratic cone constraint allows us to search over all nonnegative polynomials of a given degree. In both the univariate and multivariate cases, we consider approximate problems with linear inequality constraints. Additionally, our method can be modified slightly to include equality constraints such as mass preservation.

Error Bound Characterizations of the Conical Constraint Qualification in Convex Programming

Error Bound Characterizations of the Conical Constraint Qualification in Convex Programming

Obstacle avoidance guidance for planetary landing using convex trajectory and adaptive curvature regulation

Autonomous landing in complex and hazardous terrains is a critical stage of planetary in-situ exploration and sample-return missions. The design of the landing trajectory has to seek a balance between safety and fuel economy. Based on the theorems of convex trajectory and curvature guidance law, this paper proposes an obstacle avoidance guidance method with an adaptive curvature adjusting mechanism. The method remains the advantage in obstacle avoidance of the existed curvature guidance, and can further minimize fuel consumption by adopting a global optimization technique with a specific curvature constraint. Firstly, the nonconvex curvature constraint is transformed into a second-order cone constraint to construct a standard convex programming problem. The curvature adjustment strategy is then designed to adapt the trajectory to varying terrain conditions. By introducing the successive convex technique, the adaptive curvature guidance strategy is also suitable for small celestial body landing problems in nonlinear dynamic environments. Simulations of typical planetary landing scenarios are conducted to verify the effectiveness of the proposed method in improving safety and fuel efficiency.

Convex Performance Envelope for Minimum Lap Time Energy Management of Race Cars

The optimization of the energy management of modern hybrid-electric or fully electric race cars for minimum lap time requires a description of the vehicle dynamics performance envelope, that is, of the tires’ grip limit in corners, braking zones and during acceleration. In this paper, we present a computationally efficient performance envelope model in the form of convex constraints on the achievable longitudinal and lateral acceleration, on the assumption that the path on the track is given. The proposed acceleration limits are modeled velocity-dependent to take into account the effect of aerodynamic downforce present in many circuit race cars. The formulation as linear equality, inequality and second-order cone constraints allows to embed the model in a convex energy management optimization framework. To showcase the approach, we identify the model with data obtained from a state-of-the-art hybrid-electric Formula 1 car and present results for the Silverstone and Spa-Francorchamps circuits. The optimal energy management strategies can be evaluated with a computational time of less than 1 s. The optimal velocity profile subject to the performance envelope constraints is close to the measured one. The good agreement between the optimal solution and the measurement data shows that the proposed model captures the vehicle dynamics accurately enough for the purposes of energy management optimization.

Distributionally Robust Capacity Configuration for Energy Storage in Microgrid Considering Renewable Utilization

The energy storage plays an important role in the operation safety of the microgrid system. Appropriate capacity configuration of energy storage can improve the economy, safety, and renewable energy utilization of the microgrid. This study considers the uncertainty of renewable energy, and builds an energy storage capacity configuration (ESCC) in microgrid by using the distributionally robust optimization (DRO). This model co-optimizes energy storage planning, day-ahead scheduling, and renewable energy utilization of the microgrid, which derives the energy storage configuration strategy, balancing renewable energy utilization and operation economics of microgrid. The proposed model is a two-stage model with distributionally robust chance constrains. By applying decision rules, variable substitution, and duality techniques, this model is approximately transformed into a mixed integer programming problem with a second-order cone constraint, which can be directly solved. Experiments on the IEEE 33-bus system are carried out to verify the effectiveness and advantages of the proposed model.

On the exactness of a simple relaxation for the extended Celis–Dennis–Tapia subproblem

In this work, we study the so-called extended Celis–Dennis–Tapia subproblem that are classical trust region subproblems with an additional quadratic constraint. For this class of problems, we propose a family of relaxations, and study some of its key properties, for instance, convexity issues, exactness of the relaxation, sufficient condition for exactness. The relaxation proposed is obtained adapting to our case a methodology recently developed for trust region problems with an additional conic constraint. We also find sufficient conditions for exactness of the proposed relaxation and discuss the relations between them. Finally, some numerical experiments are presented for testing the proposed relaxation and the sufficient conditions for its tightness.

Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of [Formula: see text] is closely related to the exactness of semidefinite program (SDP) relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to [Formula: see text]. We consider the case where [Formula: see text] is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that [Formula: see text] is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature—the case of one linear matrix inequality and the S-lemma—where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit. Funding: This work was supported by the Office of Naval Research [Grant N00014-19-1-2321] with the National Science Foundation (NSF) Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-1454548] and Frank A. and Helen E. Risch Faculty Development Chair. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing, which was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through the NSF Division of Computing and Communication Foundations [Grant CCF-1740425].

Distributionally robust possibilistic optimization problems

In this paper a class of optimization problems with uncertain linear constraints is discussed. It is assumed that the constraint coefficients are random vectors whose probability distributions are only partially known. Possibility theory is used to model imprecise probabilities. In one of interpretation, a possibility distribution (a membership function of a fuzzy set) in the set of coefficient realizations induces a necessity measure, which in turn defines a family of probability distributions in this set. The distributionally robust approach is then used to transform the imprecise constraints into deterministic counterparts. Namely, the uncertain left-hand side of each constraint is replaced with the expected value with respect to the worst probability distribution that can occur. It is shown how to represent the resulting problem by using linear or second-order cone constraints. This leads to problems which are computationally tractable for a wide class of optimization models, in particular for linear programming.

Proportional–integral projected gradient method for conic optimization

Conic optimization is the minimization of a differentiable convex objective function subject to conic constraints. We propose a novel primal–dual first-order method for conic optimization, named proportional–integral projected gradient method (PIPG). PIPG ensures that both the primal–dual gap and the constraint violation converge to zero at the rate of O(1/k), where k is the number of iterations. If the objective function is strongly convex, PIPG improves the convergence rate of the primal–dual gap to O(1/k2). Further, unlike any existing first-order methods, PIPG also improves the convergence rate of the constraint violation to O(1/k3). We demonstrate the application of PIPG in constrained optimal control problems.

Power allocation in D2D enabled cellular network with probability constraints: A robust Stackelberg game approach

In this paper, we develop a Stackelberg game with a non-uniform pricing in a two-tier network environment, where multiple device-to-device (D2D) users reuse the uplink spectrum resources of the cellular user (CU). Assuming that the base station (BS) is protected by pricing the interference from D2D users, we propose a Stackelberg game framework where BS and D2D users are treated as the leader and followers respectively. Besides, the channel state information (CSI) uncertainty in the actual communication environment is considered to be suppressed the interference fluctuation effects. Specifically, the probability constraint is included to optimize the user utility. The complex probability constraint is transformed to the deterministic form of the convex second-order cone constraint by a novel method. Furthermore, a norm approximation method is proposed to reduce the computational complexity. Finally, the analytical solution of Nash equilibrium is acquired to generate the best response to the leader and all followers. Simulation results show that the proposed scheme is more effective comparing to the algorithms without the CSI probability constraint.

First- and Second-Order Optimality Conditions for Quadratically Constrained Quadratic Programming Problems

We consider a quadratic programming problem with quadratic cone constraints and an additional geometric constraint. Under suitable assumptions, we establish necessary and sufficient conditions for optimality of a KKT point and, in particular, we characterize optimality by using strong duality as a regularity condition. We consider in details the case where the feasible set is defined by two quadratic equality constraints and, finally, we analyse simultaneous diagonalizable quadratic problems, where the Hessian matrices of the involved quadratic functions are all diagonalizable by means of the same orthonormal matrix.

A bilevel conic optimization model for routing and charging of EV fleets serving long distance delivery networks

Recent unveiling of electric semi-trucks by a number of electric vehicle manufacturers indicates that part of the existing long-distance transportation fleets may soon be electrified. Operators of electric fleets will have to select travel routes considering charging station availability and cost of charging in addition to usual factors such as congestion and travel time. This requires combined modeling of transportation and electric power networks. We present such a model that considers interactions between the two networks to develop optimal routing strategies. The problem is formulated as a multi-objective bilevel conic optimization model. The upper level obtains the routing decision by minimizing a function of charging cost and travel time. The routing decision is used in the lower level that solves the AC optimal power flow model, using second order cone constraints, to determine nodal electricity prices. The model is demonstrated using a numerical problem with 24-Node transport network supported by a modified 5-Bus PJM network. The results show that our model yields optimal routes and charging strategies to meet the objectives of fleet operators. Results also indicate that the optimal routing and charging strategies of the electrified transportation fleet can support power networks to reduce nodal prices via demand response.