Convex Inequality Constraints Research Articles

A note on duality in reverse convex optimization

In this note we explore duality in reverse convex optimization with reverse convex inequality constraints. While we are examining the special case of a finite index set of inequality constraints, we are primarily interested in the general case of an arbitrary infinite index set with neither extra topological structure nor extra measure structure.

Perceptive Locomotion Through Nonlinear Model-Predictive Control

Dynamic locomotion in rough terrain requires accurate foot placement, collision avoidance, and planning of the underactuated dynamics of the system. Reliably optimizing for such motions and interactions in the presence of imperfect and often incomplete perceptive information is challenging. We present a complete perception, planning, and control pipeline, which can optimize motions for all degrees of freedom of the robot in real time. To mitigate the numerical challenges posed by the terrain, a sequence of convex inequality constraints is extracted as local approximations of foothold feasibility and embedded into an online model-predictive controller. Steppability classification, plane segmentation, and a signed distance field are precomputed per elevation map to minimize the computational effort during the optimization. A combination of multiple-shooting, real-time iteration, and a filter-based line search is used to solve the formulated problem reliably and at high rate. We validate the proposed method in scenarios with gaps, slopes, and stepping stones in simulation and experimentally on the ANYmal quadruped platform, resulting in state-of-the-art dynamic climbing.

A Projection Neural Network to Nonsmooth Constrained Pseudoconvex Optimization.

In this article, a single-layer projection neural network based on penalty function and differential inclusion is proposed to solve nonsmooth pseudoconvex optimization problems with linear equality and convex inequality constraints, and the bound constraints, such as box and sphere types, in inequality constraints are processed by projection operator. By introducing the Tikhonov-like regularization method, the proposed neural network no longer needs to calculate the exact penalty parameters. Under mild assumptions, by nonsmooth analysis, it is proved that the state solution of the proposed neural network is always bounded and globally exists, and enters the constrained feasible region in a finite time, and never escapes from this region again. Finally, the state solution converges to an optimal solution for the considered optimization problem. Compared with some other existing neural networks based on subgradients, this algorithm eliminates the dependence on the selection of the initial point, which is a neural network model with a simple structure and low calculation load. Three numerical experiments and two application examples are used to illustrate the global convergence and effectiveness of the proposed neural network.

A continuous-time neurodynamic algorithm for distributed nonconvex nonsmooth optimization problems with affine equality and nonsmooth convex inequality constraints

In this paper, a distributed nonsmooth nonconvex optimization (DNNO) problem with affine inequality and nonsmooth convex inequality constraints is studied. A continuous-time distributed neurodynamic algorithm is proposed to solve this problem. Under the assumed conditions, for any initial state, the solution of distributed neurodynamic algorithm is bounded and globally exists, and will converge to the critical point set of distributed problems in a finite time. Compared with other DNNO algorithms, distributed neurodynamic algorithm has a lower dimension and does not need to satisfy the assumption that the feasible region is bounded. Finally, a series of numerical examples are given to verify the effectiveness of the proposed algorithm.

Distributed Global Optimization for a Class of Nonconvex Optimization With Coupled Constraints

This article examines the distributed nonconvex optimization problem with structured nonconvex objective functions and coupled convex inequality constraints on static networks. A distributed continuous-time primal-dual algorithm is proposed to solve the problem. We use the canonical transformation and Lagrange multiplier method to reformulate the nonconvex optimization problem as a convex–concave saddle point computation problem, which is subsequently solved by employing the projected primal-dual subgradient method. Sufficient conditions that guarantee the global optimality of the solution generated by the proposed algorithm are provided. Numerical and application examples are presented to demonstrate the proposed algorithm.

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. In contrast, our approach does not rely on solving SDPs accurately but approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite feasible region and the interior of the semi-definite feasible region. Specifically, we first present a prototype path-following method and show its global weak* convergence to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions. Next, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with the Monteiro-Zhang scaling technique into the prototype method. We prove two-step superlinear convergence of the resulting algorithm using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.

Convexification in energy optimization of a hybrid electric propulsion system for aerial vehicles

This paper concerns the energy management of a hybrid electric propulsion system for aerial vehicles, using convex optimization. The main contribution of this paper is the proposal of a new convexification, which simplifies the formation of the convexified problem, and the proof of equality between the original problem and the convexified problem. The primary energy management is formulated from first principles and using experimental data. The convexity of the original problem is clarified via investigating the approximation to the experimental data. Then, change of variables and equality relaxation are implemented to convexify the concave constraints. The introduced variable—battery internal energy, is proposed to convexify the battery model. The relaxation of a non-affine equality yields to new convex inequality constraints. Numerical examples and forward simulations were carried out to validate the convexified problem. The first test case verifies that the convex relaxation does not sacrifice the optimality of the solution nor does the variable change lose the original bounds. Also, the optimal control from convex optimization is demonstrated to be robust to a disturbance in power demand. Comparison with the benchmark optimization—dynamic programming, shows that convex optimization achieves a minimal objective value with less fluctuation of the optimal control value. Most significant is that the convexification reduces the optimization computation time to a level compatible with implementation in practical application.

A payoff dynamics model for generalized Nash equilibrium seeking in population games

This paper studies the problem of generalized Nash equilibrium seeking in population games under general affine equality and convex inequality constraints. In particular, we design a novel payoff dynamics model to steer the decision-making agents to a generalized Nash equilibrium of the underlying game, i.e., to a self-enforceable state where the constraints are satisfied and no agent has incentives to unilaterally deviate from her selected strategy. Moreover, using Lyapunov stability theory, we provide sufficient conditions to guarantee the asymptotic stability of the corresponding equilibria set in stable population games. Auxiliary results characterizing the properties of the equilibria set are also provided for general continuous population games. Furthermore, our theoretical developments are numerically validated on a Cournot game considering various market-related and production-related constraints.

Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems

In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimum of the directional derivative at a reference point over the unit sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the unit sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving convex minimization problems (defined by directional derivatives) over the unit sphere.

Semi-global exponential stability of augmented primal–dual gradient dynamics for constrained convex optimization

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point.

Online Distributed Algorithms for Seeking Generalized Nash Equilibria in Dynamic Environments

In this article, we study the distributed generalized Nash equilibrium (GNE) seeking problem of noncooperative games in dynamic environments. Each player in the game aims to minimize its own time-varying cost function subject to a local action set. The action sets of all players are coupled through a shared convex inequality constraint. Each player can only have access to its own cost function, its own set constraint, and a local block of the inequality constraint, and can only communicate with its neighbors via a connected graph. Moreover, players do not have prior knowledge of their future cost functions. To address this problem, an online distributed algorithm is proposed based on consensus algorithms and a primal-dual strategy. The performance of the algorithm is measured by using dynamic regrets. Under mild assumptions on graphs and cost functions, we prove that if the deviation of the variational GNE sequence increases within a certain rate, then the regrets, as well as the violation of inequality constraint, grow sublinearly. A simulation is presented to demonstrate the effectiveness of our theoretical results.

An interior point sequential quadratic programming-type method for log-determinant semi-infinite programs

In this paper, we consider a nonlinear semi-infinite program that minimizes a function including a log-determinant (logdet) function over positive definite matrix constraints and infinitely many convex inequality constraints. We call this problem SIPLOG, where SIP stands for Semi-Infinite Program and LOG comes from LOG-det function. The main purpose of the paper is to develop an algorithm for efficiently computing a Karush–Kuhn–Tucker (KKT) point for the SIPLOG. More specifically, we propose an interior point sequential quadratic programming-type method that solves inexactly a sequence of semi-infinite quadratic programs approximating the SIPLOG. Furthermore, to generate a search direction in the dual matrix space associated with the semi-definite constraint, we solve scaled Newton equations that yield the family of Monteiro–Zhang directions. We prove that the proposed method weakly* converges to a KKT point under some mild assumptions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method.

Continuous-Time Algorithm for Approximate Distributed Optimization With Affine Equality and Convex Inequality Constraints

A distributed optimization problem (DOP) with affine equality and convex inequality constraints is studied in this article. First, the consensus constraint of the considered DOP is relaxed and a related approximate DOP (ADOP) is presented. It is proved that the optimal solutions of the ADOP (i.e., the near-optimal solutions of the original DOP) are able to approach the optimal solutions of the original DOP. A continuous-time algorithm is proposed for the ADOP and it is demonstrated that the state solution of the presented algorithm converges to the critical point set of the ADOP with general locally Lipschitz continuous objective functions. This means the presented algorithm is efficient for distributed nonconvex optimization problems. Particularly, when the objective functions are convex ones, the state solution of the presented algorithm is further proved to converge to a near-optimal solution of the original DOP. One illustrative example and an application on load sharing problems are shown to validate the effectiveness of the proposed algorithm.

On the convergence of inexact augmented Lagrangian methods for problems with convex constraints

We consider the augmented Lagrangian method (ALM) for constrained optimization problems in the presence of convex inequality and convex abstract constraints. We focus on the case where the Lagrangian sub-problems are solved up to approximate stationary points, with increasing accuracy. We analyze two different criteria of approximate stationarity for the sub-problems and we prove the global convergence to stationary points of ALM in both cases.

Energy Efficiency Beamforming Design for UAV Communications With Broadband Hybrid Polarization Antenna Arrays

Stringent physical resource constraints in unmanned aerial vehicle (UAV) communications bring new challenges to energy efficient transmission. In this paper, we study energy efficient beamforming for UAV communications, where a wideband hybrid polarized antenna array with tapped-delay-lines (TDLs) structure is deployed on the UAV. By exploiting hybrid structured, TDLs, and polarized antenna arrays, the strict requirements of hardware size and cost on UAV can be satisfied. It is shown that the energy efficient beamforming with the adopted array can be formulated as a WMMSE-based parameter optimization problem, which however is still a challenging nonconvex problem due to unit modulus constraints in analog beamformer. To overcome the nonconvex nature, an iterative algorithm is proposed, in which the optimization problem is iteratively solved through three convex quadratic programming problems with convex equality or inequality constraints. Finally, the numerical results demonstrate that the proposed wideband beamforming algorithm can significantly improve the energy efficiency, compared with the gradient projection algorithm.

A neurodynamic approach to nonlinear optimization problems with affine equality and convex inequality constraints

This paper presents a neurodynamic approach to nonlinear optimization problems with affine equality and convex inequality constraints. The proposed neural network endows with a time-varying auxiliary function, which can guarantee that the state of the neural network enters the feasible region in finite time and remains there thereafter. Moreover, the state with any initial point is shown to be convergent to the critical point set when the objective function is generally nonconvex. Especially, when the objective function is pseudoconvex (or convex), the state is proved to be globally convergent to an optimal solution of the considered optimization problem. Compared with other neural networks for related optimization problems, the proposed neural network in this paper has good convergence and does not depend on some additional assumptions, such as the assumption that the inequality feasible region is bounded, the assumption that the penalty parameter is sufficiently large and the assumption that the objective function is lower bounded over the equality feasible region. Finally, some numerical examples and an application in real-time data reconciliation are provided to display the well performance of the proposed neural network.

Constrained optimal consensus in multi-agent systems with single- and double-integrator dynamics

ABSTRACTThis paper fully studies distributed optimal consensus problem in undirected dynamical networks. We consider a group of networked agents that are supposed to rendezvous at the optimal point of a collective convex objective function. Each agent has no knowledge about the global objective function and only has access to its own local objective function, which is a portion of the global one, and states information of agents within its neighbourhood set. In this setup, all agents coordinate with their neighbours to seek the consensus point that minimises the network's global objective function. In the current paper, we consider agents with single-integrator and double-integrator dynamics. Further, it is supposed that agents' movements are limited by some convex inequality constraints. In order to find the optimal consensus point under the described scenario, we combine the interior-point optimisation algorithm with a consensus protocol and propose a distributed control law. The associated convergence analysis based on Lyapunov stability analysis is provided.

On Decentralized Optimization for a Class of Multisubsystem Codesign Problems

Codesign refers to the process of integrating the optimization of the physical plant design and control of a system. In this paper, a new class of codesign problems with a multisubsystem architecture in both design and control is formulated and solved. Our work here extends earlier research on models and solution approaches from single system to multisubsystem codesign. In this class, the optimization model for the physical design part in each subsystem is assumed to have a convex objective function with convex inequality and linear equality constraints. The optimization model for the control part of each subsystem belongs to a class of finite time-horizon linear quadratic regulator (LQR) feedback control. A new multilevel decentralized method is proposed that can obtain optimal or near-optimal solutions for this class of codesign problems. Details of the model and approach are presented and demonstrated by a numerical as well as a more complex spring–mass–damper system example. The proposed decentralized approach has been compared with a centralized approach. Using a scalable test problem, it is shown that as the size of the problem is increased, the computation effort for the decentralized approach increases linearly while that of the centralized approach increases nonlinearly.

A Simple Parallel Algorithm with an $O(1/t)$ Convergence Rate for General Convex Programs

This paper considers convex programs with a general (possibly nondifferentiable) convex objective function and Lipschitz continuous convex inequality constraint functions. A simple algorithm is developed and achieves an $O(1/t)$ convergence rate. Similar to the classical dual subgradient algorithm and the alternating direction method of multipliers (ADMM) algorithm, the new algorithm has a parallel implementation when the objective and constraint functions are separable. However, the new algorithm has a faster $O(1/t)$ convergence rate compared with the best known $O(1/\sqrt{t})$ convergence rate for the dual subgradient algorithm with primal averaging. Further, it can solve convex programs with nonlinear constraints, which cannot be handled by the ADMM algorithm. The new algorithm is applied to a multipath network utility maximization problem and yields a decentralized flow control algorithm with the fast $O(1/t)$ convergence rate.

Semidefinite Relaxations for MIMO Transmissions With High-Order QAM Constellations

In this letter, simple semidefinite relaxations are proposed for the detection of high-order quadrature amplitude modulations in multiple-input–multiple-output systems. The detector is based on the addition of several convex and concave quadratic inequality constraints into the maximum-likelihood detector before relaxing the problem into a convex one. Combined with a randomized sampling procedure, it is shown that the performance and worst-case computational complexity of the proposed approach makes it very competitive against existing detectors in large problem sizes.