Semi-infinite Constraints Research Articles

Robust Max-Min Fairness Transmission Design for IRS-Aided Wireless Network Considering User Location Uncertainty

In this paper, we propose a robust max-min fairness transmission design for intelligent reflecting surface (IRS)-aided wireless network in the presence of user location uncertainty. In particular, the non-isotropic reflection property for the IRS element is considered. We investigate the joint design of the active transmit beamformer at the base station (BS) and the passive phase shift matrix along with the deployment orientation (facing/pointing direction) of the IRS for maximizing the worst-case minimum signal-to-interference-plus-noise ratio (SINR) received by the users. In order to show the potential gains obtained by adjusting the deployment orientation of the IRS, a single-input-single-output (SISO) system is studied where a closed-form signal-to-noise ratio (SNR) of the user is obtained. For the multi-user case, to solve the resulting non-convex problem, an inexact-alternating-optimization algorithm consisting of a double-loop iteration is proposed. Specifically, in the inner loop, an optimization problem with semi-infinite constraints needs to be solved to increase the worst-case min-SINR compared to the given SINR reference value. We first transform the semi-infinite constraints into linear matrix inequality (LMI) constraints with finite form by applying the Taylor expansion approximation method, the general S-procedure, and the general sign-definiteness lemma. Then an efficient alternating optimization (AO) algorithm based on the two-dimensional search method, negative square penalty (NSP) method, and successive convex approximation (SCA) technique is proposed. While in the outer loop, we update the given SINR reference value as the worst-case minimum SINR obtained after each inner loop iteration. The whole algorithm terminates when the updated SINR reference values converge. Simulation results demonstrate the effectiveness of the proposed algorithm, and also show the additional system performance gain brought by the optimization of the IRS deployment orientation compared to its counterpart with fixed IRS deployment orientation, especially for a smaller IRS element number and a more prominent non-isotropic reflection property of the IRS element.

Robust Beamforming Design for RIS-Aided Cell-Free Systems With CSI Uncertainties and Capacity-Limited Backhaul

In this paper, we consider the robust beamforming design in a reconfigurable intelligent surface (RIS)-aided cell-free (CF) system considering the channel state information (CSI) uncertainties of both the direct channels and cascaded channels at the transmitter with capacity-limited backhaul. We jointly optimize the precoding at the access points (APs) and the phase shifts at multiple RISs to maximize the worst-case sum rate of the CF system subject to the constraints of maximum transmit power of APs, unit-modulus phase shifts, limited backhaul capacity, and bounded CSI errors. By applying a series of transformations, the non-smoothness and semi-infinite constraints are tackled in a low-complexity manner that facilitates the design of an alternating optimization (AO)-based iterative algorithm. The proposed algorithm divides the considered problem into two subproblems. For the RIS phase shifts optimization subproblem, we exploit the penalty convex-concave procedure (P-CCP) to obtain a stationary solution and achieve effective initialization. For precoding optimization subproblem, successive convex approximation (SCA) is adopted with a convergence guarantee to a Karush-Kuhn-Tucker (KKT) solution. Numerical results demonstrate the effectiveness of the proposed robust beamforming design, which achieves superior performance with low complexity. Moreover, the importance of RIS phase shift optimization for robustness and the advantages of distributed RISs in the CF system are further highlighted.

A Robust Adaptive Objective Power Allocation in Cognitive NOMA Networks.

Cognitive radio (CR) is a candidate for opportunistic spectrum implementation in wireless communications, allowing secondary users (SUs) to share the spectrum with primary users (PUs). In this paper, a robust adaptive target power allocation strategy for cognitive nonorthogonal multiple access (NOMA) networks is proposed, which involves the maximum transmission power of each SU and interference power threshold under PU constraints. By introducing the signal-to-interference-plus-noise ratio (SINR) adjustment factor, the strategy enables single-station communication to achieve energy efficiency (EE) or high throughput (HT), thus making the target function more flexible. In the same communication scenario, different cognitive users can choose different communication targets that meet their needs. Different QoS can be selected by the same cognitive user at different times. In the case of imperfect channel state information (CSI), semi-infinite (SI) constraints with bounded uncertainty sets are transformed into an optimization problem under the worst case, which is solved by the dual decomposition method. Simulation results show that this strategy has good adaptive selectivity and robustness.

Distributed State Fusion Estimation of Multi-Source Localization Nonlinear Systems.

For the state estimation problem of a multi-source localization nonlinear system with unknown and bounded noise, a distributed sequential ellipsoidal intersection fusion estimation algorithm based on the dual set-membership filtering method is proposed to ensure the reliability of the localization system. First, noise with unknown and bounded characteristics is modeled by using bounded ellipsoidal regions. At the same time, local estimators are designed at the sensor link nodes to filter out the noise interference in the localization system. The local estimator is designed using the dual set-membership filtering algorithm. It uses the dual principle to find the minimizing ellipsoid that can contain the nonlinear function by solving the optimization problem with semi-infinite constraints, and a first-order conditional gradient algorithm is used to solve the optimization problem with a low computational complexity. Meanwhile, the communication confusion among multiple sensors causes the problem of unknown correlation. The obtained estimates of local filters are fused at the fusion center by designing a distributed sequential ellipsoid intersection fusion estimation algorithm to obtain more accurate fusion localization results with lower computational cost. Finally, the stability and reliability of the proposed distributed fusion algorithm are verified by designing a simulation example of a multi-source nonlinear system.

Safeguarding NOMA Networks via Reconfigurable Dual-Functional Surface Under Imperfect CSI

This paper investigates the use of the reconfigurable dual-functional surface to guarantee the full-space secure transmission in non-orthogonal multiple access (NOMA) networks. In the presence of eavesdroppers, the downlink communication from the base station to the legitimate users is safeguarded by the simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS), where three practical operating protocols, namely energy splitting (ES), mode selection (MS), and time splitting (TS), are studied. The joint optimization of power allocation, active and passive beamforming is investigated to maximize the secrecy energy efficiency (SEE), taking into account the imperfect channel state information (CSI) of all channels. For ES, by approximating the semi-infinite constraints with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula> -procedure and general sign-definiteness, the problem is solved by an alternating optimization framework. Besides, the proposed algorithm is extended to the MS protocol by solving a mixed-integer non-convex problem. While for TS, a two-layer iterative method is proposed. Simulation results show that: 1) The proposed STAR-RIS assisted NOMA networks are able to provide up to 33.6% higher SEE than conventional RIS counterparts; 2) TS and ES protocols are generally preferable for low and high power domain, respectively; 3) The accuracy of CSI estimation and the bit resolution power consumption are crucial to reap the SEE benefits offered by STAR-RIS.

Tractable model discrimination for safety–critical systems with disjunctive and coupled constraints

This paper considers the model discrimination problem among a finite number of models in safety–critical systems that are subjected to constraints that can be disjunctive and where state and input constraints can be coupled with each other. In particular, we consider both the optimal input design problem for active model discrimination that is solved offline as well as the online passive model discrimination problem via a model invalidation framework. To overcome the issues associated with non-convex and generalized semi-infinite constraints due to the disjunctive and coupled constraints, we propose some techniques for reformulating these constraints in a computationally tractable manner by leveraging the Karush–Kuhn–Tucker (KKT) conditions and introducing binary variables, thus recasting the active and passive model discrimination problems into tractable mixed-integer linear/quadratic programming (MILP/MIQP) problems. When compared with existing approaches, our method is able to obtain the optimal solution and is observed in simulations to also result in less computation time. Finally, we demonstrate the effectiveness of the proposed active model discrimination approach for estimating driver intention with disjunctive safety constraints and state–input coupled curvature constraints, as well as for fault identification.

An entropic regularized method of centers for continuous minimax problem with semi infinite constraints

The purpose of this paper is to focus on continuous minimax problems with semi infinite constraints. We begin by proposing an algorithm for solving this kind of problems. The proposed algorithm combines the parametric approach and the Huard method of centers. That’s we extend the Method of Centers of Roubi for solving minimax fractional programs to continuous minimax problems. On the other hand, calculating the exact optimal solution of the proposed parametric problems, requires an extraordinary computational work per iteration. To overcome this problem we use an iterative entropic regularization method which allows us to solve each auxiliary problem inexactly generating an approximate sequence of optimal values and then the continuous minimax problem is reduced into a sequence of finite minimax problems. Examples for illustration are given to test our algorithm.

Optimal design of controlled environment agricultural systems under market uncertainty

We present a novel methodology for the simultaneous robust design and scheduling of controlled environment agricultural (CEA) systems under multi-period risk. This problem is formulated as a semi-infinite program with several semi-infinite constraints pertaining to mean-variance risk exposure with uncertain covariance over each period in the planning horizon. The general model enables robust optimization of CEA systems for cultivation of any crop portfolio under any number of cultivation modes, with a solution that represents an optimal design and operating schedule that is robust to worst-case uncertainty. Therefore, this methodology provides a conservative basis for engineering and investment decision-making and represents, to our knowledge, the first robust optimization approach to CEA systems. Our approach effectively increases the robustness of CEA systems to market uncertainty, improves the long-term economics of CEA systems over naïve operating strategies, and validates the economic viability of single and multi-mode CEA production of distinct crop portfolios.

A distributionally robust optimization approach for two-stage facility location problems

In this paper, we consider a facility location problem where customer demand constitutes considerable uncertainty, and where complete information on the distribution of the uncertainty is unavailable. We formulate the optimal decision problem as a two-stage stochastic mixed integer programming problem: an optimal selection of facility locations in the first stage and an optimal decision on the operation of each facility in the second stage. A distributionally robust optimization framework is proposed to hedge risks arising from incomplete information on the distribution of the uncertainty. Specifically, by exploiting the moment information, we construct a set of distributions which contains the true distribution and where the optimal decision is based on the worst distribution from the set. We then develop two numerical schemes for solving the distributionally robust facility location problem: a semi-infinite programming approach which exploits moments of certain reference random variables and a semi-definite programming approach which utilizes the mean and correlation of the underlying random variables describing the demand uncertainty. In the semi-infinite programming approach, we apply the well-known linear decision rule approach to the robust dual problem and then approximate the semi-infinite constraints through the conditional value at risk measure. We provide numerical tests to demonstrate the computation and properties of the robust solutions.

Robust Optimization Approach Using Scenario Concepts for Artillery Firing Scheduling Under Uncertainty

Real wars involve a considerable number of uncertainties when determining firing scheduling. This study proposes a robust optimization model that considers uncertainties in wars. In this model, parameters that are affected by enemy’s behavior and will, i.e., threats from enemy targets and threat time from enemy targets, are assumed as uncertain parameters. The robust optimization model considering these parameters is an intractable model with semi-infinite constraints. Thus, this study proposes an approach to obtain a solution by reformulating this model into a tractable problem; the approach involves developing a robust optimization model using the scenario concept and finding a solution in that model. Here, the combinations that express uncertain parameters are assumed by scenarios. This approach divides problems into master and subproblems to find a robust solution. A genetic algorithm is utilized in the master problem to overcome the complexity of global searches, thereby obtaining a solution within a reasonable time. In the subproblem, the worst scenarios for any solution are searched to find the robust solution even in cases where all scenarios have been expressed. Numerical experiments are conducted to compare robust and nominal solutions for various uncertainty levels to verify the superiority of the robust solution.

Varying confidence levels for CVaR risk measures and minimax limits

Conditional value at risk (CVaR) has been widely studied as a risk measure. In this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with CVaR appearing in the objective and also the constraints. We start by considering a problem in which CVaR is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one. We make use of a consistent tail condition which ensures that the CVaR of a random function will converge uniformly to its supremum as the confidence level increases, and establish an error bound for the CVaR optimal solution under second order growth conditions. The results are extended to a minimization problem with a constraint on the CVaR value which in the limit as the confidence level approaches one coincides with a problem having semi-infinite constraints. We study the sample average approximation scheme for the CVaR constraints and establish an exponential rate of convergence for the sample averaged optimal solution. We propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to CVaR-constrained problems out of sample. Our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance.

Robust Secure Beamforming in MIMO Wiretap Channels With Deterministically Bounded Channel Errors

In this paper, a multiple-input multiple-output wiretap channel model is studied, where the information packet sent from the transmitter to the desired receiver is wiretapped by a potential eavesdropper. When ideal channel state information (CSI) is available, we design the secure beamforming using a proposed iterative closed-form updating scheme based on majorization–minimization approach. Numerical results demonstrate that the proposed scheme acquires the same secrecy rate as the existing algorithms whereas it has much lower complexity. When ideal CSI is not available and channel errors are modeled by deterministically bounded model, we investigate robust secure beamforming (RSB) optimization problem whose goal is worst-case secrecy rate maximization. In the formulated optimization problem, the objective function includes the expression of a logarithm of a determinant ( $\log \det$ ) minus $\log \det$ . Instead of approximating the $\log \det$ expression as a trace, we propose to linearize two $\log \det$ terms. After linearization, epigraph reformulation is used to deal with deterministically bounded channel errors and semi-infinite constraints are recast as linear matrix inequalities. Finally, an alternating optimization algorithm is utilized to find the RSB design. After obtaining the RSB design, we derive the worst-case secrecy rate. Numerical results illustrate that the proposed RSB design is superior to the nonrobust one.

Decoupled beamforming techniques for distributed MIMO two-way multi-relay networks with imperfect CSI

This paper investigates decoupled beamforming techniques for the distributed multi-input multi-output (MIMO) two-way relay networks (TWRN) with imperfect channel state informations (CSIs). The objective of this paper is to maximize the weighted sum rate (SR) with semi-infinite relay power constraints. Since the objective problem is difficult to be solved directly, considering the high signal-to-residual-interference-plus-noise ratio (SRINR) and employing the Cauchy-Schwarz inequality and S-lemma, the problem can be approximately converted into a source beamforming decoupled one. In addition, with the optimal relay beamforming design and maximum ratio combining (MRC) at the receiver, the MIMO channels are decoupled into parallel single-input single-output (SISO) channels. By this way, the suboptimal relay beamforming matrix can be efficiently obtained with minimal relay power constraint. Specifically, the semi-infinite constraints can be reformulated into a linear matrix inequality (LMI), which can be efficiently solved by using an alternating optimization algorithm. Numerical results demonstrate that our proposed decoupled beamforming scheme outperforms the existing works in terms of the SR and the computational complexity with satisfactory convergence.

Distributionally Robust Reward-Risk Ratio Optimization with Moment Constraints

Reward-risk ratio optimization is an important mathematical approach in finance. We revisit the model by considering a situation where an investor does not have complete information on the distribution of the underlying uncertainty and consequently a robust action is taken to mitigate the risk arising from ambiguity of the true distribution. We consider a distributionally robust reward-risk ratio optimization model varied from the ex ante Sharpe ratio where the ambiguity set is constructed through prior moment information and the return function is not necessarily linear. We transform the robust optimization problem into a nonlinear semi-infinite programming problem through standard Lagrange dualization and then use the well-known entropic risk measure to construct an approximation of the semi-infinite constraints. We solve the latter by an implicit Dinkelbach method. Finally, we apply the proposed robust model and numerical scheme to a portfolio optimization problem and report some preliminary numerical ...

An approximation scheme for stochastic programs with second order dominance constraints

It is well known that second order dominance relation between two random variables can be described by a system of stochastic semi-infinite inequalities indexed by $\mathcal R$, the set of all real number. In this paper, we show the index set can be reduced to the support set of the dominated random variable strengthening a similar result established by Dentcheva and Ruszczynski [9] for discrete random variables. Viewing the semi-infinite constraints as an extreme robust risk measure, we relax it by replacing it with entropic risk measure and regarding the latter as an approximation of the former in an optimization problem with second order dominance constraints. To solve the entropic approximation problem, we apply the well known sample average approximation method to discretize it. Detailed analysis is given to quantify both the entropic approximation and sample average approximation for various statistical quantities including the optimal value, the optimal solutions and the stationary points obtained from solving the sample average approximated problem. The numerical scheme provides an alternative to the mainstream numerical methods for this important class of stochastic programs.

Robust unit commitment with $$n-1$$ n - 1 security criteria

The short-term unit commitment and reserve scheduling decisions are made in the face of increasing supply-side uncertainty in power systems. This has mainly been caused by a higher penetration of renewable energy generation that is encouraged and enforced by the market and policy makers. In this paper, we propose a two-stage stochastic and distributionally robust modeling framework for the unit commitment problem with supply uncertainty. Based on the availability of the information on the distribution of the random supply, we consider two specific models: (a) a moment model where the mean values of the random supply variables are known, and (b) a mixture distribution model where the true probability distribution lies within the convex hull of a finite set of known distributions. In each case, we reformulate these models through Lagrange dualization as a semi-infinite program in the former case and a one-stage stochastic program in the latter case. We solve the reformulated models using sampling method and sample average approximation, respectively. We also establish exponential rate of convergence of the optimal value when the randomization scheme is applied to discretize the semi-infinite constraints. The proposed robust unit commitment models are applied to an illustrative case study, and numerical test results are reported in comparison with the two-stage non-robust stochastic programming model.

Modeling probability densities with sums of exponentials via polynomial approximation

We propose a method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on shape-constrained optimization with exponential functions. Each function is lower and upper bounded on sub-intervals by low-degree polynomials. Thus, the constraints can be approximated with polynomial inequalities that can be implemented with linear matrix inequalities. Convexity is preserved, but the problem has now a finite number of constraints. We show how to take advantage of the properties of the exponential function in order to build quickly accurate approximations. The problem used for illustration is the least-squares fitting of a positive sum of exponentials to an empirical probability density function. When the exponents are given, the problem is convex, but we also give a procedure for optimizing the exponents. Several examples show that the method is flexible, accurate and gives better results than other methods for the investigated problems.

Robust Precoding and Beamforming Design of OSTBC Based CR AF-MIMO Relay System with Energy Harvesting Receiver

In this paper, we investigate the robust precoding and beamforming design problem for orthogonal space---time block code (OSTBC) based cognitive radio amplify-and-forward multiple-input---multiple-output relay systems with simultaneous wireless information and power transfer which consists of a primary receiver, a secondary source, a secondary relay, a secondary destination and an energy harvesting receiver. The secondary source and the secondary relay employ OSTBC based precoding and beamforming. The secondary relay and the secondary destination employ maximum-ratio combining for signal detecting and decoding. Our objective is to maximize the worst-case achievable rate under the sum transmit power constraint, the worst-case sum harvested power constraint and the worst-case interference power constraint. We transform the objective function into a convex function and convert the semi-infinite constraints into linear matrix inequalities by using $${\mathcal {S}}$$S-Lemma and the extension of $${\mathcal {S}}$$S-Lemma. Simulation results demonstrate that our proposed robust precoding and beamforming design has significant performance gain over the non-robust one.

A Cutting Surface Algorithm for Semi-Infinite Convex Programming with an Application to Moment Robust Optimization

We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as nonpolynomial moments, can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with nondifferentiable semi-infinite constraints indexed by an infinite dimensional index set. Examples comparing the cutting surface algorit...

Entropic Approximation for Mathematical Programs with Robust Equilibrium Constraints

In this paper, we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi-infinite complementarity constraints (SICC). We propose a numerical scheme for tackling SICC. Specifically, by relaxing the complementarity constraints and then randomizing the index set of SICC, we employ the well-known entropic risk measure to approximate the semi-infinite constraints with a finite number of stochastic inequality constraints. Under some moderate conditions, we quantify the approximation in terms of the feasible set and the optimal value. The approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules. Finally, we extend the discussion to a mathematical program with distributionally robust equilibrium constraints, which is essentially a one stage stochastic program with semi-infinite stochastic constraints indexed by some probability measures from an ambiguity set defined through the KL-divergence.