Convex Programming Research Articles

Max-Linear Regression by Convex Programming

Max-Linear Regression by Convex Programming

A set of linearized partially-parallel splitting methods for separable convex programs

The proximal partially-parallel splitting method (PPSM), originally proposed in Wang, Desai, and He (2017), is a hybrid mechanism that inherits the nice properties of both Gauss-Seidel and Jacobian substitution procedures for solving the multiple-block convex minimization problem, whose objective function is the sum of m individual (separable) functions without any shared variables, subject to a linear coupling constraint. In this paper, we extend this work and present some linearized versions of the PPSM, which fully utilize the separable structure and result in subproblems that either have closed-form solutions or are relatively easy to solve as compared to their original nonlinear versions. Global convergence of these linearized methods under the projection contraction algorithmic framework is proven, and furthermore, detailed remarks that serve to clarify the interconnections between these linearized variants are highlighted. Finally, the worst-case O(1/t) convergence rate of these methods under ergodic conditions is also established.

Safe Reinforcement Learning via a Model-Free Safety Certifier.

This article presents a data-driven safe reinforcement learning (RL) algorithm for discrete-time nonlinear systems. A data-driven safety certifier is designed to intervene with the actions of the RL agent to ensure both safety and stability of its actions. This is in sharp contrast to existing model-based safety certifiers that can result in convergence to an undesired equilibrium point or conservative interventions that jeopardize the performance of the RL agent. To this end, the proposed method directly learns a robust safety certifier while completely bypassing the identification of the system model. The nonlinear system is modeled using linear parameter varying (LPV) systems with polytopic disturbances. To prevent the requirement for learning an explicit model of the LPV system, data-based λ -contractivity conditions are first provided for the closed-loop system to enforce robust invariance of a prespecified polyhedral safe set and the system's asymptotic stability. These conditions are then leveraged to directly learn a robust data-based gain-scheduling controller by solving a convex program. A significant advantage of the proposed direct safe learning over model-based certifiers is that it completely resolves conflicts between safety and stability requirements while assuring convergence to the desired equilibrium point. Data-based safety certification conditions are then provided using Minkowski functions. They are then used to seemingly integrate the learned backup safe gain-scheduling controller with the RL controller. Finally, we provide a simulation example to verify the effectiveness of the proposed approach.

On the number of pivots of Dantzig's simplex methods for linear and convex quadratic programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegatively constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency of simplex-type methods for solving these problems. Two applications of the quadratic programming results are presented.

Computation‐efficiency‐oriented resource optimization in NOMA‐assisted multi‐UAV‐enabled MEC networks

SummaryUnmanned aerial vehicles (UAVs) can serve as aerial mobile edge computing (MEC) servers to provide computing services to Internet of Things smart devices (ISDs) with insufficient computing capacity on the ground. However, how to provide energy‐efficient computing services by designing appropriate resource optimization strategy is still a challenge issue in UAV‐enabled MEC networks. To this end, this paper proposes a computation efficiency (CE)‐oriented partial offloading framework for UAV‐enabled MEC networks, where the ISDs' computation bits and energy consumption are taken into account simultaneously. Specifically, we firstly divide the ISDs into multiple clusters and determine the deployment of UAVs by k‐means method. Meanwhile, ISDs occupying the same subchannel in the same cluster can offload data through non‐orthogonal multiple access (NOMA) technology. Then, the problem of maximizing the CE of the system is formulated by optimizing subchannel allocation, transmit power and computation resources of ISDs. To solve it, we propose a staged optimization approach by using matching theory, Dinkelbach and sequential convex programming (SCP) methods. Numerical results demonstrate the proposed scheme can achieve higher CE compared with other baseline schemes.

Virtual Quantum Resource Distillation.

Distillation, or purification, is central to the practical use of quantum resources in noisy settings often encountered in quantum communication and computation. Conventionally, distillation requires using some restricted "free" operations to convert a noisy state into one that approximates a desired pure state. Here, we propose to relax this setting by only requiring the approximation of the measurement statistics of a target pure state, which allows for additional classical postprocessing of the measurement outcomes. We show that this extended scenario, which we call "virtual resource distillation," provides considerable advantages over standard notions of distillation, allowing for the purification of noisy states from which no resources can be distilled conventionally. We show that general states can be virtually distilled with a cost (measurement overhead) that is inversely proportional to the amount of existing resource, and we develop methods to efficiently estimate such cost via convex and semidefinite programming, giving several computable bounds. We consider applications to coherence, entanglement, and magic distillation, and an explicit example in quantum teleportation (distributed quantum computing). This work opens a new avenue for investigating generalized ways to manipulate quantum resources.

Imperceptible Attacks on Fault Detection and Diagnosis Systems in Smart Buildings

Automated Fault Detection and Diagnosis (AFDD) systems are critical to safe and efficient operation of smart buildings. A significant amount of building data can be collected and analyzed to detect building compo-nent failures. Attacks against such data that are contami-nated with small additive disturbances (i.e., adversarial per-turbation attacks) could dreadfully impact the performance of such systems while maintaining a high level of imper-ceptibility. The vulnerability studies of such data attacks is lacking. Specifically, most existing detection and clas-sification models have flat structures, regarded as Single-Stage Classifiers (SSCs), are prone to adversarial data perturbation attacks. In this paper, we present a coarse-to-fine Hierarchical Fault Detection and multi-level Diagnosis (HFDD) model, and formulate a mathematical program to derive targeted attacks on the model with respect to a pre-specified target diagnosis level. Two algorithms are developed based on convex relaxations of the formulated program for non-targeted attacks. An Alternating Direction Method of Multipliers (ADMM)-based solver is developed for the convex programs. Extensive experiments are con-ducted using two real-world datasets of measurements from air handling units and chillers, demonstrating the fea-sibility of the proposed attacks with regard to misclassifi-cation rate and imperceptibility of the attack. We also show that the HFDD is more robust to disturbances than SSC-based fault detection and multi-level diagnosis systems.

Scalable Computation of Robust Control Invariant Sets of Nonlinear Systems

Ensuring robust constraint satisfaction for an infinite time horizon is a challenging, yet crucial task when deploying safety-critical systems. We address this issue by synthesizing robust control invariant sets of perturbed nonlinear sampled-data systems. This task can be encoded as a non-convex program that we approximate by a tailored, computationally efficient successive convexification algorithm. Based on the zonotopic representation of invariant sets, we obtain an updated candidate for the invariant set and the invariance-enforcing controller by solving a single convex program. To obtain a possibly large region of safe operation, our algorithm is designed so that the sequence of candidate invariant sets is volume-wise monotonically increasing. We demonstrate the efficacy and scalability of our approach using a broad range of nonlinear control systems from the literature with up to 20 dimensions.

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

This study considers equilibrium problems, focusing on identifying finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set. This comprehensive concept encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of the existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.

A Relaxed Version of the Cutting Method with Approximation of the Constraint Region

A cutting method was proposed for solving the convex programming problem. The method assumes that the constraint region of the problem is embedded into some polyhedral sets for constructing iteration points. It involves the construction of a sequence of approximations that belongs to the admissible set and is relaxed, as well as implies that the ε-solution of the initial problem is fixed after a finite number of steps. The method also allows to obtain mixed convergent algorithms by using, if desired, any known or new relaxation algorithms for constructing the main iteration points.

A new sensitivity-based robust optimisation of structures under bounded type uncertainty

This paper deals with robust design optimisation (RDO) of structures when uncertainty information about system parameters is limited. For such parameters, neither the statistical moments, nor the probability distribution is available. Rather, based on practical considerations, only bounds of variations of such parameters under uncertainty can be set, which is in-turn used to model them as uncertain-but-bounded (UBB) type. In recent years, convex programming (CP) approaches are used to solve RDO problem with UBB parameters. However, these approaches generally do not consider sensitivity information while formulating an RDO problem. However, it is well accepted that sensitivity is the key focus of an RDO as by definition a robust design should be least sensitive to uncertainty effects. Thus, a new sensitivity-importance-based RDO formulation is proposed in this paper, where a new sensitivity index is defined and minimised along with the usual cost function subjected to a dispersion-related constraint. Then, the proposed RDO is solved through usual CP framework. The improvement by the proposed approach is demonstrated taking four application examples. The results indicate that the proposed approach yields more robust solutions compared to the conventional approaches.

A cutting-plane method with internal iteration points for the general convex programming problem

A cutting method for solving the problem of convex programming was proposed. The method calculates iteration points based on approximation by polyhedral sets of the constraint region and the epigraph of the objective function. Its distinguishing feature is that the main sequence of approximations is constructed within the admissible region. At each step, it is also possible to assess how close the current value of the function is to the optimal value. The convergence of the method was proved. A few of its implementations were outlined.

Secure state estimation via robust optimization for nonlinear cyber‐physical systems

AbstractThis article proposes secure state estimation for cyber‐physical systems against sensor attacks. The attack and defense strategies are established via additional historical data, and the defender aims to reduce the estimation error maximally while the attacker aims to degrade the system performance maximally. The algorithm is implemented in the Nash equilibrium framework where the defender first designs the defense strategy and then the attacker designs corresponding attack parameters to launch attacks. Then, a robust optimization problem is formulated using Wasserstein ambiguity sets, which turn out to be equivalent to a convex program. A novel secure observer is proposed, where the attack estimation is used to mitigate attacks. Moreover, the detector is to monitor system behavior and detects the existence of sensor attacks. Finally, simulation results and comparative results illustrate the effectiveness of the defense strategy.

Evaluation of Grid Support Capacity From Electrified Railways

High power level and temporal operation flexibility endow electric trains with great potential for improving grid resilience. It becomes crucial to quantify the grid support capacity (GSC) from large-scale electrified railways. This paper innovatively proposes a hierarchical train coordination and trajectory regulation method to evaluate and deliver the GSC from electrified railways. In the light of the mechanical energy consumption analysis of train running, the energy-oriented train operation model is proposed to avoid the nonlinear differential dynamics of train kinetics when coordinating large-scale trains. Then the aggregation and disaggregation of trains for grid support can be formulated as linear and convex quadratic programming problems respectively. To closely follow the reference power and enhance the feasibility of train operation, a novel iterative discretization approach is proposed to generate the precise reference trajectory. A sliding mode controller with disturbance bound adaption is devised to overcome the modeling errors and disturbances when deploying GSC in realistic environment. Comprehensive numerical examples verify the effectiveness of the proposed scheme, and demonstrate the significant grid support potential from collaborative electrified railways.

A steepest feasible direction method for linear programming. Derivation and embedding in the simplex method

A feasible direction method for linear programming has been proposed. The method is embedded in the framework of the simplex method, even though it works with non-edge feasible directions. The direction used is the steepest in the space of all variables or an approximation thereof, and it is found by solving a strictly convex quadratic program in the space of the nonbasic variables. Further, this program guarantees the feasibility of the direction even in the case of degeneracy. To remain within the simplex framework, the direction is represented by an auxiliary, or external, nonbasic column, which is a nonnegative linear combination of original nonbasic columns. We have made an experimental evaluation of the suggested method on both nondegenerate and highly degenerate problem instances. The overall results are very promising for continued research along this line, especially concerning various computational strategies that can be applied when the method is implemented.

МЕТОДЫ НЕЛИНЕЙНОГО ПРОГРАММИРОВАНИЯ В РЕШЕНИИ КОНЕЧНЫХ ИГР

The paper considers the formulation of nonlinear programming problems for finding a Nash equilibrium situation in a bimatric game and in a game with three participants. An overview of existing methods for solving is given: an algorithm for solving bimatric LemkeHowson games, methods for solving quadratic and convex programming problems for a finite game with three participants.

A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming

A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming

Risk-Averse Decision Making Under Uncertainty

A large class of decision making under uncertainty problems can be described via Markov decision processes (MDPs) or partially observable MDPs (POMDPs), with application to artificial intelligence and operations research, among others. In this paper, we consider the problem of designing policies for MDPs and POMDPs with objectives and constraints in terms of dynamic coherent risk measures rather than the traditional total expectation, which we refer to as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constrained risk-averse problem</i> . Our contributions can be described as follows: For MDPs, under some mild assumptions, we propose an optimization-based method to synthesize Markovian policies. We then demonstrate that such policies can be found by solving difference convex programs (DCPs). We show that our formulation generalize linear programs for constrained MDPs with total discounted expected costs and constraints; For POMDPs, we show that, if the coherent risk measures can be defined as a Markov risk transition mapping, an infinite-dimensional optimization can be used to design Markovian belief-based policies. For POMDPs with stochastic finite-state controllers (FSCs), we show that the latter optimization simplifies to a (finite-dimensional) DCP. We incorporate these DCPs in a policy iteration algorithm to design risk-averse FSCs for POMDPs. We demonstrate the efficacy of the proposed method with numerical experiments involving conditional-value-at-risk (CVaR) and entropic-value-at-risk (EVaR) risk measures.

Two-stage robust programming model for active distribution network with a high proportion of renewable energy

To reduce operating costs and obtain the optimal location and capacity of renewable energy access, a two-stage robust planning model is studied for active distribution networks with a high proportion of renewable energy access. To build a function to minimize annual comprehensive costs, the constraints of power flow, branch power, and node voltage of the active distribution network are studied. Considering the characteristics and properties of power flow safety constraints, the mixed integer second-order cone transformation is used to construct a mixed integer convex programming model, which is convenient for efficiently obtaining the global optimal solution. To avoid the influence of uncertainty, a two-stage robust model is constructed. The explicit maximum value method is used to obtain the solution results that satisfy the planning model of all uncertain problems. Experiments show that the model can obtain the optimal access position and capacity of renewable energy. After application, it can reduce the annual comprehensive cost of the distribution network and ensure the stable and safe operation of the active distribution network.

Dispersion Sensitive Optimal Control: A Conditional Value-at-Risk-Based Tail Flattening Approach via Sequential Convex Programming

Dispersion Sensitive Optimal Control: A Conditional Value-at-Risk-Based Tail Flattening Approach via Sequential Convex Programming