Convex Constraints Research Articles

Utility Maximization Under Trading Constraints with Discontinuous Utility

This paper investigates a utility maximization problem in a Black--Scholes market, in which trading is subject to a convex cone constraint and the utility function is not necessarily continuous or ...

Efficient Computation of User Optimal Traffic Assignment via Second-Order Cone and Linear Programming Techniques

Static traffic assignment aims to disclose the spatial distribution of vehicular flow over a transportation network subject to given traffic demands, and plays an essential role in transportation engineering. User-optimal pattern adheres the individual rationality of motorists, in which everyone chooses a route that minimizes his own travel cost, while considering congestion effects influenced by the aggregated movement of vehicles. User optimal traffic assignment, which is also known as the user equilibrium, entails solving an optimization problem with a strictly convex objective function and linear constraints. However, the performances of general-purpose solvers are quite disappointing. This paper proposes two highly efficient computation models for the user equilibrium problem. The first one exploits the second-order cone reformulation of a convex power function, resulting in a second-order cone program, and no approximation is incurred. The second one approximates the convex objective function using a piece-wise linear function, and comes down to a linear program. An adaptive path generation oracle is devised in order to circumvent path enumeration in problem setup. Case studies demonstrate that the proposed method can deal with large-scale transportation systems, and outperforms the most popular iterative algorithm in the literature.

Distributed Subgradient-Free Stochastic Optimization Algorithm for Nonsmooth Convex Functions over Time-Varying Networks

In this paper we consider a distributed stochastic optimization problem without gradient/subgradient information for local objective functions and subject to local convex constraints. Objective functions may be nonsmooth and observed with stochastic noises, and the network for the distributed design is time-varying. By adding stochastic dithers to local objective functions and constructing randomized differences motivated by the Kiefer--Wolfowitz algorithm, we propose a distributed subgradient-free algorithm for finding the global minimizer with local observations. Moreover, we prove that the consensus of estimates and global minimization can be achieved with probability one over the time-varying network, and we obtain the convergence rate of the mean average of estimates as well. Finally, we give numerical examples to illustrate the performance of the proposed algorithms.

Projected Stochastic Gradients for Convex Constrained Problems in Hilbert Spaces

Convergence of a projected stochastic gradient algorithm is demonstrated for convex objective functionals with convex constraint sets in Hilbert spaces. In the convex case, the sequence of iterates...

Августиновская пропускная способность и августиновский центр

For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set. The Augustin mean is shown to be the unique fixed point of an operator defined in terms of the order and the input distribution. The Augustin information is shown to be continuously differentiable in the order. For any channel and convex constraint set with finite Augustin capacity, the existence of a unique Augustin center and the associated van Erven-Harremoes bound are established. The Augustin-Legendre (A-L) information, capacity, center, and radius are introduced and the latter three are proved to be equal to the corresponding Renyi-Gallager quantities. The equality of the A-L capacity to the A-L radius for arbitrary channels and the existence of a unique A-L center for channels with finite A-L capacity are established. For all interior points of the feasible set of cost constraints, the cost constrained Augustin capacity and center are expressed in terms of the A-L capacity and center. Certain shift invariant families of probabilities and certain Gaussian channels are analyzed as examples.

Network localisation using Lagrangian optimisation

The network localisation problem with non-convex distance constraints may be modelled as a nonlinear optimisation problem. The existing localisation techniques either eliminate the nonconvex constraints or relax them into convex constraints to employ the traditional convex optimisations like semi-definite programming (SDP), least square approximation, etc. to find the node positions. We propose a method to solve the original network localisation problem with noisy distance measurements without any modification of non-convex constraints. Using the nonlinear Lagrangian technique for non-convex optimisation, we convert the localisation problem to a root finding problem involving single variable. This problem is then solved using bisection method. For computing functional values, it involves finite mini-max problem (FMX). We use sequential quadratic programming (SQP) to fix the FMX problem. Simulations studies show that, the number of iterations in the proposed method is reasonable to achieve any desired label of accuracy in node positions.

Fitting heavy-tailed mixture models with CVaR constraints

Abstract Standard methods of fitting finite mixture models take into account the majority of observations in the center of the distribution. This paper considers the case where the decision maker wants to make sure that the tail of the fitted distribution is at least as heavy as the tail of the empirical distribution. For instance, in nuclear engineering, where probability of exceedance (POE) needs to be estimated, it is important to fit correctly tails of the distributions. The goal of this paper is to supplement the standard methodology and to assure an appropriate heaviness of the fitted tails. We consider a new Conditional Value-at-Risk (CVaR) distance between distributions, that is a convex function with respect to weights of the mixture. We have conducted a case study demonstrating e˚ciency of the approach. Weights of mixture are found by minimizing CVaR distance between the mixture and the empirical distribution. We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution.

General risk measures for robust machine learning

A wide array of machine learning problems are formulated as the minimization of the expectation of a convex loss function on some parameter space. Since the probability distribution of the data of interest is usually unknown, it is is often estimated from training sets, which may lead to poor out-of-sample performance. In this work, we bring new insights in this problem by using the framework which has been developed in quantitative finance for risk measures. We show that the original min-max problem can be recast as a convex minimization problem under suitable assumptions. We discuss several important examples of robust formulations, in particular by defining ambiguity sets based on $ \varphi $-divergences and the Wasserstein metric. We also propose an efficient algorithm for solving the corresponding convex optimization problems involving complex convex constraints. Through simulation examples, we demonstrate that this algorithm scales well on real data sets.

Wireless Information and Power Transfer in Single User OFDM Systems

In this article, OFDM-based single-user simultaneous wireless information and power transfer (SWIPT) system for power splitting (PS) and time switching (TS) plans is analyzed. The optimal design for resource allocation (i.e., power allocation and subcarrier assignment) is investigated in order to maximize the sum rate with a minimum transferred power constraint. By extending power allocation and time division to more spans, two new schemes TPS2 and TPS1 which are obtained by appropriate combination of TS and PS will be analyzed and simulated. In proposed method, we will divide the time frame of data transfer into more subintervals that are equipped with power split hardware. The problem of transferring information in this system becomes a constraint convex optimization problem. By solving this non-convex optimization problem with mathematical methods, increase in the rate of information transfer can be found comparing to the conventional schemes. Results are presented as the rate of information transfer in terms of different values of the minimum required harvested energy. Simulation results show that the proposed method, i.e. PTS2, has about 11% superiority in comparison with conventional PS and TS schemes in the data transfer rate for the same energy harvesting value. Also in this method, with the increase in the number of sub-intervals an ideal simultaneous transmission can be achieved, in which energy harvesting will not lead to reduction in any data transfer rate.

Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

Robust control of systems with sector nonlinearities via convex optimization: A data‐driven approach

SummaryIn this paper, a new data‐driven method for designing robust controllers is proposed for systems with sector‐bounded nonlinearities and multimodel uncertainties. The results from the circle criterion are used to generate necessary and sufficient convex constraints that guarantee the stability of the closed‐loop system. The main feature of the proposed approach is that only the frequency response data of the linear part of the system is used for guaranteeing the stability of the closed‐loop nonlinear system. Additionally, a convex optimization problem is formulated to ensure performance with respect to the fundamental component of a sector‐bounded nonlinearity. The case study illustrates how the proposed method can be used to control uncertain systems that are subject to sector‐bounded nonlinearities.

Weighted Network Design With Cardinality Constraints via Alternating Direction Method of Multipliers

This paper examines simultaneous design of the network topology and the corresponding edge weights in the presence of a cardinality constraint on the edge set. Network properties of interest for this design problem lead to optimization formulations with convex objectives, convex constraint sets, and cardinality constraints. This class of problems is referred to as the cardinality-constrained optimization problem (CCOP); the cardinality constraint generally makes CCOPs NP-hard. In this paper, a customized alternating direction method of multipliers (ADMM) algorithm aiming to improve the scalability of the solution strategy for large-scale CCOPs is proposed. This algorithm utilizes the special structure of the problem formulation to obtain closed-form solutions during each iterative step of the corresponding ADMM updates. We also provide a convergence proof of the proposed customized ADMM to a stationary point under certain conditions. Simulation results illustrate that the customized ADMM algorithm has a significant computational advantage over existing methods, particularly for large-scale network design problems.

An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints

In this paper, based on the hyperplane projection technique, we propose a three-term conjugate gradient method for solving nonlinear monotone equations with convex constraints. Due to the derivative-free feature and lower storage requirement, the proposed method can be applied to the solution of large-scale non-smooth nonlinear monotone equations. Under some mild assumptions, the global convergence is proved when the line search fulfils the backtracking line search condition. Moreover, we prove that the proposed method is R-linearly convergent. Numerical results show that our method is competitive and efficient for solving large-scale nonlinear monotone equations with convex constraints.

Limited descent-based mean value method for inverse reliability analysis

The robustness and efficiency of inverse reliability methods are important issues in reliability-based design optimization (RBDO) using performance measure approach (PMA). The adaptive modified chaos control (ACC), step length adjustment (SA), and relaxed mean value (RMV) methods were recently implemented to improve the robustness and efficiency of PMA. In this paper, a limited descent mean value (LDMV) method is proposed to improve robustness and efficiency of inverse reliability analysis for either convex or concave probabilistic constraints. The LDMV formula is dynamically adjusted by an adaptive step size based on the advanced mean value method (AMV). The robustness and efficiency of the ACC, SA, RMV, and proposed LDMV methods are compared through six nonlinear performance functions. The results illustrated a similar robust performance between LDMV against RMV and FSL methods and superior to the ACC method. The proposed LDMV improves the robustness and efficiency of the inverse first-order reliability method in comparison with existing reliability methods.

Solving Reverse Convex Programming Problems Using a Generalized Cutting-Plane Method

It is well known that each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is supremally generated by affine functions. More precisely, each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is the upper envelope of its affine minorants. In this paper, we propose an algorithm for solving reverse convex programming problems by using such a representation together with a generalized cutting-plane method. Indeed, by applying this representation, we solve a sequence of problems with a smaller feasible set, in which the reverse convex constraint is replaced by a still reverse convex but polyhedral constraint. Moreover, we prove that the proposed algorithm converges, under suitable assumptions, to an optimal solution of the original problem. This algorithm is coded in MATLAB language and is evaluated by some numerical examples.

Bare Bones Teaching Learning-Based Optimization Technique for Economic Emission Load Dispatch Problem Considering Transmission Losses

Economic emission load dispatch plays a great role in power system cost analysis. The purpose of economic emission load dispatch is to minimize the fuel and emission cost satisfying load demand. Various iterative techniques had been used to solve economic emission load dispatch problem in previous years by different authors. In this article, the new teaching learning-based optimization (TLBO) technique using variant bare bones TLBO has been proposed for solving economic emission load dispatch problem with convex and non-convex constraints by considering transmission losses and valve point loading effects. Conventional TLBO has two phases like teacher phase and learner phase. In teacher phase, learners update their knowledge through sharing knowledge with teacher, and finally, learners improve their knowledge by interactions among learners. Bare bones TLBO employs an interactive learning strategy, which is the hybridization of the learning strategy in the conventional TLBO. The proposed algorithm has been applied on six-, ten- and forty-unit systems, and the results obtained are compared with existing techniques; hence, superiority of the proposed algorithm is proved.

On $$\epsilon $$ ϵ -solutions for robust semi-infinite optimization problems

In this paper, we consider semi-infinite optimization problems involving a convex objective function and infinitely many convex constraint functions with data uncertainty, and give its robust counterpart \(\mathrm{(RSIP)}\). Moreover, we consider approximate solutions (\(\epsilon \)-solutions) for \(\mathrm{(RSIP)}\). Using robust optimization approach (worst-case approach), we establish robust necessary optimality and robust sufficient theorems and give duality results for \(\epsilon \)-solutions for \(\mathrm{(RSIP)}\) under a closed convex cone constraint qualification. Moreover, an example is given to illustrate the obtained duality results.

Low-Rank Optimization With Convex Constraints

The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design, and low-complexity modeling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this nonconvex problem is convexified by nuclear-norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original nonconvex problem coincide. A semidefinite programming representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear-norm regularization and may outperform balanced truncation.

Reducing the Number of Elements in the Synthesis of a Broadband Linear Array With Multiple Simultaneous Frequency-Invariant Beam Patterns

The problem of reducing the number of elements in a broadband linear array with multiple simultaneous crossover frequency-invariant (FI) patterns is considered. Different from the single FI pattern array case, every element channel in the multiple FI pattern array is divided and followed by multiple finite-impulse-response (FIR) filters, and each of the multiple FIR filters has a set of coefficients. In this situation, a collective filter coefficient vector and its energy bound are introduced for each element, and then the problem of reducing the number of elements is transformed as minimizing the number of active collective filter coefficient vectors. In addition, the radiation characteristics including beam pointing direction, mainlobe FI property, sidelobe level, and space-frequency notching requirement for each of the multiple patterns can be formulated as multiple convex constraints. The whole synthesis method is implemented by performing an iterative second-order cone programming (SOCP). This method can be considered as a significant extension of the original SOCP for synthesizing broadband sparse array with single FI pattern. Numerical synthesis results show that the proposed method by synthesizing multiple discretized crossover FI patterns can save more elements than the original iterative SOCP by using a single continuously scannable FI pattern for covering the same space range. Moreover, even for multiple FI-patterns case with complicated space-frequency notching, the proposed method is still effective in the reduction of the number of elements.

Partially symmetrical derivative-free Liu–Storey projection method for convex constrained equations

ABSTRACTApplying Powell symmetrical technique to the Liu–Storey conjugate gradient method, a partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search. By using some line searches, the global convergence is proved merely by assuming that the equations are Lipschitz continuous. Moreover, we prove the R-linear convergence rate of the proposed method with an additional assumption. Finally, compared with one existing method, the performance of the proposed method is showed by some numerical experiments on the given test problems.