Convex Cost Function Research Articles

Push–Pull Gradient Methods for Distributed Optimization in Networks

In this article, we focus on solving a distributed convex optimization problem in a network, where each agent has its own convex cost function and the goal is to minimize the sum of the agents' cost functions while obeying the network connectivity structure. In order to minimize the sum of the cost functions, we consider new distributed gradient-based methods where each node maintains two estimates, namely an estimate of the optimal decision variable and an estimate of the gradient for the average of the agents' objective functions. From the viewpoint of an agent, the information about the gradients is pushed to the neighbors, whereas the information about the decision variable is pulled from the neighbors, hence giving the name “push-pull gradient methods.” The methods utilize two different graphs for the information exchange among agents and, as such, unify the algorithms with different types of distributed architecture, including decentralized (peer to peer), centralized (master-slave), and semicentralized (leader-follower) architectures. We show that the proposed algorithms and their many variants converge linearly for strongly convex and smooth objective functions over a network (possibly with unidirectional data links) in both synchronous and asynchronous random-gossip settings. In particular, under the random-gossip setting, “push-pull” is the first class of algorithms for distributed optimization over directed graphs. Moreover, we numerically evaluate our proposed algorithms in both scenarios, and show that they outperform other existing linearly convergent schemes, especially for ill-conditioned problems and networks that are not well balanced.

Fast value iteration: an application of Legendre-Fenchel duality to a class of deterministic dynamic programming problems in discrete time

ABSTRACTWe propose an algorithm, which we call ‘Fast Value Iteration’ (FVI), to compute the value function of a deterministic infinite-horizon dynamic programming problem in discrete time. FVI is an efficient algorithm applicable to a class of multidimensional dynamic programming problems with concave return (or convex cost) functions and linear constraints. In this algorithm, a sequence of functions is generated starting from the zero function by repeatedly applying a simple algebraic rule involving the Legendre-Fenchel transform of the return function. The resulting sequence is guaranteed to converge, and the Legendre-Fenchel transform of the limiting function coincides with the value function.

A Nesterov-Like Gradient Tracking Algorithm for Distributed Optimization Over Directed Networks

In this article, we concentrate on dealing with the distributed optimization problem over a directed network, where each unit possesses its own convex cost function and the principal target is to minimize a global cost function (formulated by the average of all local cost functions) while obeying the network connectivity structure. Most of the existing methods, such as push-sum strategy, have eliminated the unbalancedness induced by the directed network via utilizing column-stochastic weights, which may be infeasible if the distributed implementation requires each unit to gain access to (at least) its out-degree information. In contrast, to be suitable for the directed networks with row-stochastic weights, we propose a new directed distributed Nesterov-like gradient tracking algorithm, named as D-DNGT, that incorporates the gradient tracking into the distributed Nesterov method with momentum terms and employs nonuniform step-sizes. D-DNGT extends a number of outstanding consensus algorithms over strongly connected directed networks. The implementation of D-DNGT is straightforward if each unit locally chooses a suitable step-size and privately regulates the weights on information that acquires from in-neighbors. If the largest step-size and the maximum momentum coefficient are positive and small sufficiently, we can prove that D-DNGT converges linearly to the optimal solution provided that the cost functions are smooth and strongly convex. We provide numerical experiments to confirm the findings in this article and contrast D-DNGT with recently proposed distributed optimization approaches.

Collusion-proof Mechanisms for Multi-Unit Procurement

A principal wants to procure a discrete number of homogeneous units from a finite set of agents. Each agent has an increasing and convex cost function, whose exact shape is unknown to the principal. Utility is quasilinear in money. We characterize the set of mechanisms that are strategy-proof and robust to collusion. The latter is captured by reallocation-proofness (group strategy-proofness) if the colluding agents can (cannot) make side deals among each other. We show that in every reallocation-proof mechanism, the principal offers the agents a fixed price per unit. While group-strategy-proof mechanisms can be more complex, they cannot guarantee that the principal will receive exactly her desired quantity. We fully characterize the set of group-strategy-proof mechanisms with a uniform price.

Push-Pull Finite-Time Convergence Distributed Optimization Algorithm

With the widespread application of distributed systems, many problems need to be solved urgently. How to design distributed optimization strategies has become a research hotspot. This article focuses on the solution rate of the distributed convex optimization algorithm. Each agent in the network has its own convex cost function. We consider a gradient-based distributed method and use a push-pull gradient algorithm to minimize the total cost function. Inspired by the current multi-agent consensus cooperation protocol for distributed convex optimization algorithm, a distributed convex optimization algorithm with finite time convergence is proposed and studied. In the end, based on a fixed undirected distributed network topology, a fast convergent distributed cooperative learning method based on a linear parameterized neural network is proposed, which is different from the existing distributed convex optimization algorithms that can achieve exponential convergence. The algorithm can achieve finite-time convergence. The convergence of the algorithm can be guaranteed by the Lyapunov method. The corresponding simulation examples also show the effectiveness of the algorithm intuitively. Compared with other algorithms, this algorithm is competitive.

Should a Sharing Platform Adopt the Bilateral Review System?

The sharing economy and platforms have gained popularity in recent years, raising concerns about demand-side information asymmetry: the cost to serve each buyer is typically unknown to sellers. To mitigate this concern, some platforms like Airbnb adopt the bilateral review system (BRS), where both buyers and sellers can rate each other, different from the traditional unilateral review system (URS). We examine the effect of different review systems on the operation of sharing platforms. We analyze a decentralized platform in which heterogeneous sellers with one-unit capacity set prices simultaneously and then get matched with buyers with different serving costs. We show that with URS, even when perfect seller information is provided, the high-quality sellers can still be driven out of the market, owing to buyer information asymmetry and the “co-production” nature of the serving cost, in contrast to adverse selection in the classic used-car market. Additionally, we identify the critical role of serving cost structure. Although BRS can alleviate the adverse selection problem with a convex cost function and benefits the platform, it is detrimental to the platform with a linear cost function. Interestingly, we find that providing buyer information could help buyers but hurt sellers. Moreover, BRS could lead to a win-win-win outcome for the platform, buyers, and sellers. Our results not only shed light on the review system design but also provide a plausible explanation for the business practice that BRS is widely used by sharing platforms like Airbnb and Fiverr.

Determination of the optimal option for production of the convex cost function

The choice of the optimal option for the production of in the construction of buildings and structures is based on a preliminary generalization and analysis of the most promising options for the implementation of construction and installation from which the most rational optimal in the is selected conditions under consideration. The purpose of the study is to determine the optimal option for production of at the convex cost function. We used the method of the determination of costs on the duration of works, represented by a convex function showing the shorter the duration of the project the more costs are required when it is reduced by the same amount. The key parameters affecting the final result of construction are the duration cost and quality of performed. Consequently, one of the main tasks is to draw up a works schedule with close coordination of the resources necessary for their implementation. To do this we have to solve problems associated with the distribution of limited resources. In this case, the technological interdependence between the works performed, which may be of a different nature, should be taken into account. As the results, new approaches were founded to developing a strategy for the selection of options for construction and installation works in order to ensure contractual deadlines for their implementation and minimize the funds spent on their implementation taking into account the nature of the dependencies between the works performed is an urgent and important problem.

Bellman equations for scalar linear convex stochastic control problems

A family of discrete time stochastic control problems with linear dynamics and convex cost functionals are studied. For the case of a scalar control for such a model with additive finite time horizon, discounted, and average cost per unit time convex cost

Distributed Dual Gradient Tracking for Resource Allocation in Unbalanced Networks

This paper proposes a distributed dual gradient tracking algorithm (DDGT) to solve resource allocation problems over an unbalanced network, where each node in the network holds a private cost function and computes the optimal resource by interacting only with its neighboring nodes. Our key idea is the novel use of the distributed push-pull gradient algorithm (PPG) to solve the dual problem of the resource allocation problem. To study the convergence of the DDGT, we first establish the sublinear convergence rate of PPG for non-convex objective functions, which advances the existing results on PPG as they require the strong-convexity of objective functions. Then we show that the DDGT converges linearly for strongly convex and Lipschitz smooth cost functions, and sublinearly without the Lipschitz smoothness. Finally, experimental results suggest that DDGT outperforms existing algorithms.

Distributed Online Convex Optimization With Time-Varying Coupled Inequality Constraints

This paper considers distributed online optimization with time-varying coupled inequality constraints. The global objective function is composed of local convex cost and regularization functions and the coupled constraint function is the sum of local convex functions. A distributed online primal-dual dynamic mirror descent algorithm is proposed to solve this problem, where the local cost, regularization, and constraint functions are held privately and revealed only after each time slot. Without assuming Slater's condition, we first derive regret and constraint violation bounds for the algorithm and show how they depend on the stepsize sequences, the accumulated dynamic variation of the comparator sequence, the number of agents, and the network connectivity. As a result, under some natural decreasing stepsize sequences, we prove that the algorithm achieves sublinear dynamic regret and constraint violation if the accumulated dynamic variation of the optimal sequence also grows sublinearly. We also prove that the algorithm achieves sublinear static regret and constraint violation under mild conditions. Assuming Slater's condition, we show that the algorithm achieves smaller bounds on the constraint violation. In addition, smaller bounds on the static regret are achieved when the objective function is strongly convex. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

A fatigue-oriented cost function for optimal individual pitch control of wind turbines

In a context of wind power production growth, it is necessary to optimize the levelized cost of energy by maximizing the profit of operating wind turbines, which is defined as the difference between gains (e.g. energy sold) and losses (e.g. operation and maintenance cost). Operation and maintenance cost is intimately related to fatigue damage. However, considering fatigue directly in an optimization needs to be carefully done because its faithful model does not fit standard forms. In this paper, it is first shown that the variance of a signal and its corresponding damage using fatigue theory are nonlinearly related. Therefore, this relationship is used to design a convex cost function approximating fatigue. Preliminary tests suggest promising results regarding the relevance of this formulation in optimizing fatigue trade-off when compared to a family of quadratic cost functions. The proposed formulation allows to directly consider economic parameters in the cost function, limiting thus the sensitivity to parameter tuning.

Passivity-based generalization of primal–dual dynamics for non-strictly convex cost functions

In this paper, we revisit primal–dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. We hypothesize that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function. Then, we prove this hypothesis based on the passivity paradigm together with the invariance principle for Carathéodory systems. Additionally, we show that the presented algorithm is also a generalization of existing augmented Lagrangian-based primal–dual dynamics and discuss the benefit of the presented generalization in terms of noise reduction and convergence speed. Finally, the presented algorithm is demonstrated through simulation of lighting control in a building.

Sparse Recovery and Dictionary Learning From Nonlinear Compressive Measurements

Sparse coding and dictionary learning are popular techniques for linear inverse problems such as denoising or inpainting. However in many cases, the measurement process is nonlinear, for example for clipped, quantized or 1-bit measurements. These problems have often been addressed by solving constrained sparse coding problems, which can be difficult to solve, and assuming that the sparsifying dictionary is known and fixed. Here we propose a simple and unified framework to deal with nonlinear measurements. We propose a cost function that minimizes the distance to a convex feasibility set, which models our knowledge about the nonlinear measurement. This provides an unconstrained, convex, and differentiable cost function that is simple to optimize, and generalizes the linear least squares cost commonly used in sparse coding. We then propose proximal based sparse coding and dictionary learning algorithms, that are able to learn directly from nonlinearly corrupted signals. We show how the proposed framework and algorithms can be applied to clipped, quantized and 1-bit data.

Exact spectral-like gradient method for distributed optimization

Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where n nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we propose a novel exact distributed gradient method wherein nodes’ step-sizes are designed according to the novel rules akin to those in spectral gradient methods. We refer to the proposed method as Distributed Spectral Gradient method. The method exhibits R-linear convergence under standard assumptions for the nodes’ local costs and safeguarding on the algorithm step-sizes. We illustrate the method’s performance through simulation examples.

Continuous-Time Integral Dynamics for a Class of Aggregative Games With Coupling Constraints

We consider continuous-time equilibrium seeking in a class of aggregative games with strongly convex cost functions and affine coupling constraints. We propose simple semidecentralized integral dynamics and prove their global asymptotic convergence to a variational generalized aggregative or Nash equilibrium. The proof is based on Lyapunov arguments and invariance techniques for differential inclusions.

The Effective Cooperative Diffusion Strategies With Adaptation Ability by Learning Across Adaptive Network-Wide Systems

In this paper, we consider the nonlinear least squares (NLLSs) problems in adaptive networks, where a collection of nodes with adaptation ability by learning are required to estimate a global vector parameter by minimizing the specified convex cost function. Although the global Gauss-Newton (GN) method is an excellent candidate for solving such problems, many challenges need to be addressed for practical realization due to its natures of centralization and noncooperation. Without specialized design for routing, we motivate and propose new diffusion GN methods with cooperative strategy among local neighborhoods. The good performances of diffusion cooperation schemes have been proved in different literatures, such as distribution, robustness, and easy implementation. The proposed cooperative diffusion strategies are named as aggregation-then-update (ATU) and update-then-aggregation (UTA), which reach fully information diffusion across network and consist of two steps in a reversible way including aggregation of local estimates and local GN update. Although all implementations are local, the cooperation between nodes is network wide. Based on the steady-state equilibria theory in the nonlinear discrete dynamical system, the convergence analysis of proposed algorithms is provided. The results show that the global convergence can be achieved when the sufficient conditions are satisfied. We also provide performance comparisons and analysis together with simulation to confirm the applicability and effectiveness of proposed diffusion GN algorithms.

Adaptive online distributed optimization in dynamic environments

ABSTRACT This paper presents a dynamic regret analysis on the decentralized online convex optimization problems computed over a network of agents. The goal is to distributively optimize a global function which can be decomposed into the summation of local cost functions associated with each agent. By exploiting the convexity of cost functions, previous studies have shown that the dynamic regret of a decentralized gradient-based algorithm can be upper bounded by the path-length of the comparator sequence and the connectivity in the underlying network. In this paper, we illustrate that the dynamic regret can be further improved by allowing the learner to use a class of adaptive search directions and step-sizes from estimates of first and second moments of the gradients. Specifically, we develop a new class of decentralized and stochastic algorithms based on the adaptive gradient method in the dynamic environment and show that their regret bounds for certain realistic classes of loss functions are considerably better than existing bounds. Numerical simulations verify the theoretical results and demonstrate the efficiency of the new proposed method in practice.

Differentially Private Mechanism for Determining the Equilibrium Price under Strongly Convex Cost Functions and Strongly Concave Utility Functions

Differentially Private Mechanism for Determining the Equilibrium Price under Strongly Convex Cost Functions and Strongly Concave Utility Functions

Generalizations of weighted matroid congestion games: pure Nash equilibrium, sensitivity analysis, and discrete convex function

Congestion games provide a model of human’s behavior of choosing an optimal strategy while avoiding congestion. In the past decade, matroid congestion games have been actively studied and their good properties have been revealed. In most of the previous work, the cost functions are assumed to be univariate or bivariate. In this paper, we discuss generalizations of matroid congestion games in which the cost functions are n-variate, where n is the number of players. First, motivated from polymatroid congestion games with $$\mathrm {M}^\natural $$ -convex cost functions, we conduct sensitivity analysis for separable $$\mathrm {M}^\natural $$ -convex optimization, which extends that for separable convex optimization over base polyhedra by Harks et al. (SIAM J Optim 28:2222–2245, 2018. https://doi.org/10.1137/16M1107450 ). Second, we prove the existence of pure Nash equilibria in matroid congestion games with monotone cost functions, which extends that for weighted matroid congestion games by Ackermann et al. (Theor Comput Sci 410(17):1552–1563, 2009. https://doi.org/10.1016/j.tcs.2008.12.035 ). Finally, we prove the existence of pure Nash equilibria in matroid resource buying games with submodular cost functions, which extends that for matroid resource buying games with marginally nonincreasing cost functions by Harks and Peis (Algorithmica 70(3):493–512, 2014. https://doi.org/10.1007/s00453-014-9876-6 ).

Online Optimization Using Zeroth Order Oracles

This letter considers the iterative numerical optimization of time-varying cost functions where no gradient information is available at each iteration. In this case, the proposed algorithm estimates a directional derivative by finite differences. The main contributions are the derivation of error bounds for such algorithms and proposal of optimal algorithm parameter values, e.g., step-sizes, for strongly convex cost functions. The algorithm is applied to tackle a source localization problem using a sensing agent where the source actively evades the agent. Numerical examples are provided to illustrate the theoretical results.