Concave Maximization Research Articles

Double machine learning and design in batch adaptive experiments

Abstract We consider an experiment with at least two stages or batches and O ( N ) O\left(N) subjects per batch. First, we propose a semiparametric treatment effect estimator that efficiently pools information across the batches, and we show that it asymptotically dominates alternatives that aggregate single batch estimates. Then, we consider the design problem of learning propensity scores for assigning treatment in the later batches of the experiment to maximize the asymptotic precision of this estimator. For two common causal estimands, we estimate this precision using observations from previous batches, and then solve a finite-dimensional concave maximization problem to adaptively learn flexible propensity scores that converge to suitably defined optima in each batch at rate O p ( N − 1 ⁄ 4 ) {O}_{p}\left({N}^{-1/4}) . By extending the framework of double machine learning, we show this rate suffices for our pooled estimator to attain the targeted precision after each batch, as long as nuisance function estimates converge at rate o p ( N − 1 ⁄ 4 ) {o}_{p}\left({N}^{-1/4}) . These relatively weak rate requirements enable the investigator to avoid the common practice of discretizing the covariate space for design and estimation in batch adaptive experiments while maintaining the advantages of pooling. Our numerical study shows that such discretization often leads to substantial asymptotic and finite sample precision losses outweighing any gains from design.

A duality and free boundary approach to adverse selection

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.

Robust Versions of the Lower and Upper Possibilistic Mean - Variance Models for the One Period or Two Periods Cases

It is easy to use possibility theory in modeling incomplete information. Robust optimization is an important tool when there is parameter uncertainty. Thus, in this study, we propose robust versions of the lower and upper possibilistic mean - variance (MV) models when there are multiple possibility distribution scenarios. Here, we use entropy as a diversification constraint. In addition, we reduce these robust versions to concave maximization problems. Furthermore, we generalize them for two periods portfolio selection problem by using fuzzy addition and multiplication. On the other hand, these generalizations are not concave maximization problems. Finally, we give an illustrative example by using different solvers in Gams modeling system.

Production Planning with Risk Hedging Under a Conditional Value at Risk Objective

Effective management of demand risk is a central problem of production planning. In many applications, it is observed that demand often depends on certain asset prices in the financial market. This calls for a model that jointly optimizes the production decision along with a risk hedging policy, using the latter to compensate for loss of profit in scenarios of weak demand. In “Production Planning with Risk Hedging under a Conditional Value at Risk Objective,” Wang and Yao develop an integrated production and risk hedging model, where the risk hedging strategy is derived in explicit form and the production quantity is solved as a concave maximization problem. Furthermore, a complete characterization of the efficient frontier is provided, and the improvement over the production-only decision is quantified.

Joint Precoder and Beamformer Design for Secure Relay Networks With Finite-Alphabet Inputs and Statistical CSI of Eve

Because of discrepant deteriorations of the intended receiver and the unintended receiver, artificial noise (AN) can be invoked in conjunction with the precoder to wireless transmissions for enhancing the secrecy rate (SR) performance as long as we elaborately frame them. This paper studies a secure transmission strategy by jointly designing the precoder and AN beamformer at the relay network, where a passive eavesdropper and finite-alphabet inputs are taken into account. We propose a pair of solutions for low-order modulation and high-order modulation, respectively. To solve the first optimization problem, we propose a low-complexity algorithm with the aid of the invoked cut-off rate. Interestingly, we find that the phase of an optimum precoder for maximizing the SR has a correlation to both the channels spanning from the transmitter to the relays and spanning from the relays to the legitimate receiver. Furthermore, we reveal that the AN beamformer vector has at most one non-zero component, which only locates at the position corresponding to the minimum element of the channel between the relays and the intended receiver. According to these findings, the SR maximization problem over the two vectors is simplified as one only related to a pair of scalars. As for the high-order modulation, a new solution is further proposed for circumventing the predicament that the computational complexity exponentially increases as the size of the adopted modulation increases, where we not only eliminate two-layer summation over the legitimate symbols but also conceive a concave maximization SR problem. Finally, simulation results demonstrate the efficiency of the proposed algorithms in terms of the SR performance.

Certified and accurate SDP bounds for the ACOPF problem

We propose a new method for improving the bound tightness of the popular semidefinite programming (SDP) relaxation for the ACOPF introduced in Lavaei and Low (2012), Molzahn and Hiskens (2019). First, we reformulate the ACOPF Lagrangian dual as an unconstrained concave maximization problem with a clique decomposition induced sparse structure. We prove that this new formulation has the same optimal value as the SDP relaxation. We then use the solution of the SDP relaxation as a starting point for a tailored structure-aware bundle method. This post-processing technique significantly improves the tightness of the SDP bounds computed by the state-of-the-art solver MOSEK, as shown by our computational experiments on large-scale instances from PGLib-OPF v21.07. For ten of the tested instances, our post-processing decreases by more than 50% the optimality gap obtained with MOSEK.

A Generative Model for Texture Synthesis based on Optimal Transport Between Feature Distributions

We propose GOTEX, a general framework for texture synthesis by optimization that constrains the statistical distribution of local features. While our model encompasses several existing texture models, we focus on the case where the comparison between feature distributions relies on optimal transport distances. We show that the semi-dual formulation of optimal transport allows to control the distribution of various possible features, even if these features live in a high-dimensional space. We then study the resulting minimax optimization problem, which corresponds to a Wasserstein generative model, for which the inner concave maximization problem can be solved with standard stochastic gradient methods. The alternate optimization algorithm is shown to be versatile in terms of applications, features and architecture; in particular, it allows to produce high-quality synthesized textures with different sets of features. We analyze the results obtained by constraining the distribution of patches or the distribution of responses to a pre-learned VGG neural network. We show that the patch representation can retrieve the desired textural aspect in a more precise manner. We also provide a detailed comparison with state-of-the-art texture synthesis methods. The GOTEX model based on patch features is also adapted to texture inpainting and texture interpolation. Finally, we show how to use our framework to learn a feed-forward neural network that can synthesize on-the-fly new textures of arbitrary size in a very fast manner. Experimental results and comparisons with the mainstream methods from the literature illustrate the relevance of the generative models learned with GOTEX.

Градиентный метод с неточным оракулом для задач композитной невыпуклой оптимизации

In this paper, we develop new first-order method for composite non-convex minimization problems with simple constraints and inexact oracle. The objective function is given as a sum of `hard', possibly non-convex part, and `simple' convex part. Informally speaking, oracle inexactness means that, for the `hard' part, at any point we can approximately calculate the value of the function and construct a quadratic function, which approximately bounds this function from above. We give several examples of such inexactness: smooth non-convex functions with inexact Holder-continuous gradient, functions given by auxiliary uniformly concave maximization problem, which can be solved only approximately. For the introduced class of problems, we propose a gradient-type method, which allows to use different proximal setup to adapt to geometry of the feasible set, adaptively chooses controlled oracle error, allows for inexact proximal mapping. We provide convergence rate for our method in terms of the norm of generalized gradient mapping and show that, in the case of inexact Holder-continuous gradient, our method is universal with respect to Holder parameters of the problem. Finally, in a particular case, we show that small value of the norm of generalized gradient mapping at a point means that a necessary condition of local minimum approximately holds at that point.

Relaxed solutions for incompressible inviscid flows: a variational and gravitational approximation to the initial value problem.

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain [Formula: see text] are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation that generates a set of 'relaxed', 'multi-stream', version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an attempt to get a more relevant set of relaxed Euler equations, we address the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model, for which we show that the initial value problem can be stated as a concave maximization problem from which we can at least recover a large class of smooth solutions for short enough times. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Partial least trimmed squares regression

Partial least squares (PLS) regression is a linear regression technique and plays an important role in dealing with high-dimensional regressors. Unfortunately, PLS is sensitive to outliers in datasets and consequentially produces a corrupted model. In this paper, we propose a robust method for PLS based on the idea of least trimmed squares (LTS), in which the objective is to minimize the sum of the smallest h squared residuals. However, solving an LTS problem is generally NP-hard. Inspired by the complementary idea of Sim and Hartley, we solve the inverse of the LTS problem instead and formulate it as a concave maximization problem, which is convex and can be solved in polynomial time. Classic PLS as well as two of the most efficient robust PLS methods, Partial Robust M (PRM) regression and RSIMPLS, are compared in this study. Results of both simulation and real data sets show the effectiveness and robustness of our approach.

Designing experiments informed by observational studies

AbstractThe increasing availability of passively observed data has yielded a growing interest in “data fusion” methods, which involve merging data from observational and experimental sources to draw causal conclusions. Such methods often require a precarious tradeoff between the unknown bias in the observational dataset and the often-large variance in the experimental dataset. We propose an alternative approach, which avoids this tradeoff: rather than using observational data for inference, we use it to design a more efficient experiment. We consider the case of a stratified experiment with a binary outcome and suppose pilot estimates for the stratum potential outcome variances can be obtained from the observational study. We extend existing results to generate confidence sets for these variances, while accounting for the possibility of unmeasured confounding. Then, we pose the experimental design problem as a regret minimization problem subject to the constraints imposed by our confidence sets. We show that this problem can be converted into a concave maximization and solved using conventional methods. Finally, we demonstrate the practical utility of our methods using data from the Women’s Health Initiative.

Hybrid Precoding Design for Secure Generalized Spatial Modulation With Finite-Alphabet Inputs

Technically, the security performance of generalized spatial modulation (GenSM) networks can be enhanced by dynamically adjusting the precoder allocated to the legitimate signal as communication channel varies. For this purpose, our paper proposes a secure transmission strategy upon designing both digital and analog precoders for hybrid GenSM systems, where an eavesdropper is taken into account. The concept of the hybrid GenSM system has arose to improve the spatial multiplexing (SMX) gain for remedying the shortcoming of the limited number of radio frequency chains in traditional GenSM systems. However, this may lead to a great deal of security degradation since the SMX gain of the unintended receiver will be also improved. To this end, we develop a secrecy enhancement scheme by devising both analog and digital precoders for hybrid GenSM networks. Specifically, we derive an efficiently closed-form alternative to the original secrecy rate (SR) expression for reducing the excessive computational complexity of the joint optimization problem over the hybrid precoder. Then, by using this alternative as our cost function, an iterative algorithm is proposed. In particular, we elaborately conceive a pair of concave maximization problems in order to optimize the digital and analog precoders, respectively. Our proposed strategy not only utilizes semi-positive definite relaxing technique over the analog precoder but also invokes a lower bound of the alternative to further simplify the optimization over the vectored digital precoder. Subsequently, both the convergence and computational complexity of the proposed alternating iteration algorithm are analyzed. Compared to existing designs, our proposed strategy strikes a compelling role in balancing the SR performance and complexity. Finally, our simulation results confirm the efficiency of the proposed algorithm in terms of the SR performance achieved.

Joint relay assignment and energy‐efficiency maximization in energy‐harvesting downlink/uplink clustered nonorthogonal multiple‐access networks

AbstractIn this article, the problem of joint relay assignment and energy‐efficiency maximization (J‐RA‐EE‐MAX) in energy‐harvesting downlink (DL) and uplink (UL) clustered nonorthogonal multiple‐access (NOMA) networks is considered. Specifically, the aim is to perform relay assignment to user clusters in the DL and UL directions, while simultaneously maximizing energy efficiency (EE) over each relay via multiobjective optimization, and satisfying users' quality‐of‐service (QoS) constraints. However, problem J‐RA‐EE‐MAX happens to be nonconvex in each link direction, and thus is computationally prohibitive. Alternatively, a low‐complexity solution procedure is devised to solve problem J‐RA‐EE‐MAX in each link direction by: (i) optimally solving the energy‐efficiency maximizing power allocation (EE‐MAX‐PA) for each (user cluster, relay) combination to construct the relays' preference profile, and (ii) performing relay assignment via Gale's top trading cycles (TTC) matching mechanism. In particular, the optimal solution of the EE‐MAX‐PA problem is obtained by transforming it into a concave maximization problem, while the TTC mechanism is executed in linear time‐complexity so as to obtain a stable relay assignment. Simulation results are presented to validate the proposed solution procedure, which is shown to yield comparable user cluster sum‐rate and relay EE to the J‐RA‐EE‐MAX scheme, and superior to other schemes in both link directions, however, at lower computational complexity, while satisfying users' QoS constraints. Finally, this work sheds light on the importance of decoupling relay assignment in the DL/UL directions, which significantly improves user cluster sum‐rate and relay EE in comparison to coupled relay assignment schemes, and thus serves the requirements of 5G networks and beyond.

OPTIMAL PORTFOLIO SELECTION BY CVAR BASED SHARPE RATIO — GENETIC ALGORITHM APPROACH —

We address an optimal portfolio selection problem of maximizing so we call CVaR (Conditional Value-at-Risk) based Sharpe ratio of return rate of portfolio, which is defined as the ratio of the expected excess return to CVaR. The Sharpe ratio defined as the ratio of expected excess return to standard deviation, the most common traditional performance measure, takes standard deviation as a risk measure, however, its has been received a lot of criticisms. In our CVaR based Sharpe ratio, the standard deviation is replaced by CVaR, which is a remarkable coherent risk measure which overcomes essential defects of standard deviation. Although our new performance measure is expected to enlarge the applicable area of practical investment problems for which the original Sharpe ratio is not suitable, however, we should device effective computational methods to solve optimal portfolio selection problems with very large number of investment opportunities. In order to deal with rather complicated non-concave objective function, which comes from the introduction of CVaR, in this paper, we propose a Genetic Algorithm (GA) approach. The paper briefly reviews the literature in the area of application of Genetic Algorithms to financial problems, and then details the development of Genetic Algorithm for portfolio selection. In order to evaluate CVaR for each portfolio, by utilizing the results of Rockafellar and Uryasev (2000), we introduce an auxiliary decision variable to obtain a tractable concave maximization problem. Furthermore, if we estimate or approximate required expected values by sampling methods or historical data, we can reduce this concave maximization problem to an LP (Linear Programming) problem. Therefore, our problem could be solved by GA which incorporates LP for evaluating values of CVaR. Numerical experiments from real Japanese financial data are conducted to test our approach to conclude that, by suitable choice of probability parameter of three evolution operation, we could effectively solve optimal selection problems with practical sizes.

An Anti-Eavesdropping Strategy for Precoding-Aided Spatial Modulation With Rough CSI of Eve

In this paper, an anti-eavesdropping strategy is proposed for secure precoding-aided spatial modulation networks, under the assumption that the rough channel state information of eavesdropper can be obtained at the transmitter. Traditionally, artificial noise (AN) can be always projected into the null-space of the legitimate channel, however it may lead to some security loss since this strategy dispenses with a holistic consideration for secure transmissions. To reduce the computational complexity of our optimization problem, we derive a closed-form expression that is a loose bound of the approximate rate over the illegitimate channel. Then a concave maximization problem is formulated for optimizing the covariance matrix of AN. Simulation results show that our proposed low-complexity scheme performs closely to the method which directly maximizes the approximate secrecy rate expression, and harvests significant secrecy rate gains compared with the traditional null-space projection benchmark.

Production Planning with Risk Hedging under a CVaR Objective

A central problem in planning production capacity is how to effectively manage demand risk. We develop a model that integrates capacity planning and risk hedging decisions under a popular risk measure, conditional value at risk (CVaR). The CVaR objective generalizes the usual risk-neutral objective (such as the expected payoff), and allows for explicit modeling of the degree of aversion to downside risk (associated with low demand). The starting point of our model is to incorporate the impact on demand from a financial asset (including, for instance, a tradable market index as a proxy for the general economy) via a demand rate function. This way, in addition to the capacity decision at the beginning of the planning horizon, there is also a dynamic hedging strategy throughout the horizon, the latter plays the role of both mitigating demand risk and supplementing the payoff. The hedging strategy is restricted to partial information along with a cap on loss (pathwise). To find the optimal hedging strategy, we construct and solve a dual problem to derive the optimal terminal wealth from hedging; the real-time hedging strategy is then mapped out via the martingale representation theorem. With the hedging strategy optimized, we show that optimizing the production quantity is a concave maximization problem. With both production and hedging (jointly) optimized, we provide a complete characterization of the efficient frontier, and quantify the improvement over the production-only approach. Furthermore, via sensitivity and asymptotic analyses, we spell out the impacts of the hedging budget and the risk aversion level, along with other qualitative insights.

Adaptive Resource Allocation for Secure Two-Hop Relaying Communication

In this paper, we develop novel transmission schemes for secure dual-hop Alice–Ray–Bob relaying communication over fading channels in the presence of a passive eavesdropper (Eve). To control the risk of secrecy outage under unknown eavesdropper channel conditions, we impose secrecy constraint in terms of maximum allowable secrecy outage probability. We study the throughput–optimal buffer-aided adaptive relaying problem for two scenarios: 1) fixed (Alice and Ray) power allocation and 2) adaptive power allocation. The resulting constrained optimization problems are solved using Lagrangian approach and convex optimization. In each frame, either Alice or Ray or neither is scheduled for transmission depending on the main (Alice–Ray and Ray–Bob) channel conditions. Since the transmission schemes can result in unboundedly large (queuing) delay at Ray’s buffer, we next study the transmission schemes guaranteeing the bounded average delay. The optimal transmission problem is formulated as an infinite horizon average reward constrained Markov decision process. Subsequently, by relying on a novel state value function approach, we show that in each frame, the solution can be obtained by solving a concave maximization problem, taking into account both the main channel conditions and the buffer state. An online transmission algorithm is developed to iteratively update the state value function, which converges to the optimal solution without requiring a-priori statistical information on the fading channels. The simulation results demonstrate the effectiveness of the proposed schemes over benchmark schemes under various secrecy constraints and signal-to-noise power ratio regimes.

The Initial Value Problem for the Euler Equations of Incompressible Fluids Viewed as a Concave Maximization Problem

We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem.We show that this problem, which shares a similar structure with the optimal transport problemwith quadratic cost, in its "Benamou-Brenier" formulation,always admits a relaxed solution that can be interpretedin terms of $sub-solution$ of the Euler equations in the sense of convex integration theory.Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.

Congestion patterns of electric vehicles with limited battery capacity.

The path choice behavior of battery electric vehicle (BEV) drivers is influenced by the lack of public charging stations, limited battery capacity, range anxiety and long battery charging time. This paper investigates the congestion/flow pattern captured by stochastic user equilibrium (SUE) traffic assignment problem in transportation networks with BEVs, where the BEV paths are restricted by their battery capacities. The BEV energy consumption is assumed to be a linear function of path length and path travel time, which addresses both path distance limit problem and road congestion effect. A mathematical programming model is proposed for the path-based SUE traffic assignment where the path cost is the sum of the corresponding link costs and a path specific out-of-energy penalty. We then apply the convergent Lagrangian dual method to transform the original problem into a concave maximization problem and develop a customized gradient projection algorithm to solve it. A column generation procedure is incorporated to generate the path set. Finally, two numerical examples are presented to demonstrate the applicability of the proposed model and the solution algorithm.

Scene Classification Based on the Fully Sparse Semantic Topic Model

In high spatial resolution (HSR) imagery scene classification, it is a challenging task to recognize the high-level semantics from a large volume of complex HSR images. The probabilistic topic model (PTM), which focuses on modeling topics, has been proposed to bridge the so-called semantic gap. Conventional PTMs usually model the images with a dense semantic representation and, in general, one topic space is generated for all the different features. However, this approach fails to consider the sparsity of the semantic representation, the classification quality, as well as the time consumption. In this paper, to solve the above problems, a fully sparse semantic topic model (FSSTM) framework is proposed for HSR imagery scene classification. FSSTM, with an elaborately designed modeling procedure, is able to represent the image with sparse but representative semantics. Based on this framework, the topic weights of multiple features are exploited by solving a concave maximization problem, which improves the fusion of the discriminative semantic information at the topic level. Meanwhile, the sparsity and representativeness of the topics generated by FSSTM guarantee that the image is adaptive to the change of a topic number. FSSTM can consistently achieve a good performance with a limited number of training samples, and is robust for HSR image scene classification. The experimental results obtained with three different types of HSR image data sets confirm that the proposed algorithm is effective in improving the performance of scene classification, and is highly efficient in discovering the semantics of HSR images when compared with the state-of-the-art PTM methods.