Convex Cost Function Research Articles

An accelerated distributed online gradient push‐sum algorithm on time‐varying directed networks

AbstractThis paper investigates a distributed online optimization problem with convex objective functions on time‐varying directed networks, where each agent holds its own convex cost function and the goal is to cooperatively minimize the sum of the local cost functions. To tackle such optimization problems, an accelerated distributed online gradient push‐sum algorithm is firstly proposed, which combines the momentum acceleration technique and the push‐sum strategy. Then, we specifically analyze the regret for the proposed algorithm. The theoretical result shows that the individual regret of the proposed algorithm achieves a sublinear regret with order of , where T is the time horizon. Moreover, we implement the proposed algorithm in sensor networks for solving the distributed online estimation problem, and the results illustrate the effectiveness of the proposed algorithm.

Parametric Computation of Minimum-Cost Flows with Piecewise Quadratic Costs

We develop algorithms solving parametric flow problems with separable, continuous, piecewise quadratic, and strictly convex cost functions. The parameter to be considered is a common multiplier on the demand of all nodes. Our algorithms compute a family of flows that are each feasible for the respective demand and minimize the costs among the feasible flows for that demand. For single commodity networks with homogenous cost functions, our algorithm requires one matrix multiplication for the initialization, a rank 1 update for each nondegenerate step and the solution of a convex quadratic program for each degenerate step. For nonhomogeneous cost functions, the initialization requires the solution of a convex quadratic program instead. For multi-commodity networks, both the initialization and every step of the algorithm require the solution of a convex program. As each step is mirrored by a breakpoint in the output this yields output-polynomial algorithms in every case.

A Scalable Markov Chain Framework for Influence Maximization in Arbitrary Networks

We consider a Markov chain model of influence in a network where inactive nodes can be activated by their active in-neighbors or by external factors. The activation time is a random variable whose parameters, or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">influence rates</i> , depend on control variables whose values can be changed by the decision-maker (DM) seeking to allocate her limited budget to minimize the time until all nodes are active. We show that the optimization of either the expectation or the tail probabilities of the network activation time (NAT) is an intractable (EXPTIME) problem. Thus, we focus on a stochastic upper bound on the NAT and show that both its expectation and tail probabilities are minimized when the Markov chain's minimum sojourn rate is maximized. We provide performance guarantees for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">network activation time optimization problem</i> (NATP) with respect to the optimization of the expectation problem and show that the NATP becomes an efficiently solvable convex optimization problem when we have concave influence rates and a convex cost function. We also provide an efficient cut generation algorithm, which can solve many synthetic and real-life instances with more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^4$</tex-math></inline-formula> nodes and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^5$</tex-math></inline-formula> arcs in times ranging from a few seconds to a few minutes.

Location and two-echelon inventory network design with economies and diseconomies of scale in facility operating costs

Most existing facility location and network design models assume the facility cost as either a fixed cost term or a linear/concave function of the volume of demand assigned to a facility. In particular, a concave cost function implies decreasing marginal cost and exhibits economies of scale. However, when the assigned demand volume exceeds a certain level, diseconomies of scale may occur due to facility congestions, overuse of resources, increasing operational complexity, and so on, leading to a convex cost function with increasing marginal cost. This paper studies a warehouse–retailer network design problem, which simultaneously optimizes the warehouse locations, the warehouse–retailer assignment, and the multi-echelon inventory replenishment policy to minimize the total cost, including the warehouse location and operating costs, the inventory replenishment cost, and the transportation cost. In this paper, the warehouse operating cost is defined as a general function of the volume of demand assigned to a warehouse, which could be concave first due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation algorithm to solve its linear relaxation. The pricing problem of the column generation process, although formulated as a nonlinear mixed integer program, has nice structural properties and can be solved efficiently by a branch-and-bound method. To validate the effectiveness and efficiency of the proposed column generation algorithm, we conduct numerical studies based on randomly generated instances. Numerical results also demonstrate the impact of economies and diseconomies of scale on network design decisions and warehouse operating cost.

Distributed Adaptive Optimization With Weight-Balancing

This article addresses the continuous-time distributed optimization of a strictly convex summation-separable cost function with possibly nonconvex local functions over strongly connected digraphs. Distributed optimization methods in the literature require convexity of local functions, or balanced weights, or vanishing step sizes, or algebraic information (eigenvalues or eigenvectors) of the Laplacian matrix. The solution proposed here covers both weight-balanced and unbalanced digraphs in a unified way, without any of the aforementioned requirements.

Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system

&lt;p style="text-indent:20px;"&gt;This paper investigates the mean-field stochastic linear quadratic optimal control problem of Markov regime switching system (M-MF-SLQ, for short). The representation of the cost functional for the M-MF-SLQ is derived using the technique of operators. It is shown that the convexity of the cost functional is necessary for the finiteness of the M-MF-SLQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. We demonstrate with a few examples that our results can be employed for tackling some financial problems such as mean-variance portfolio selection problem.&lt;/p&gt;

Large-scale dynamic system optimization using dual decomposition method with approximate dynamic programming

In this paper, multi-agent dynamic optimization with a coupling constraint is studied. The aim is to minimize a strongly convex social cost function, by considering a linear stochastic dynamics for each agent and also coupling constraints among the agents. In order to handle the coupling constraint and also, to avoid high computational cost imposed by a centralized method for large scale systems, the dual decomposition method is used to decompose the problem into multiple individual sub-problems, while the dual variable is adjusted by a coordinator. Nevertheless, since each sub-problem is not a linear–quadratic (LQ) optimal control problem, and hence its closed-form solution does not exist, approximate dynamic programming (ADP) is utilized to solve the sub-problems. The main contribution of the paper is to propose an algorithm by considering the interrelated iterations of dual variable adjustment and ADP, and to prove the convergence of the algorithm to the global optimal solution of the social cost function. Additionally, the implementation of the proposed algorithm using a neural network is presented. Also, the computational advantage of the proposed algorithm in comparison with other bench-marking methods is discussed in simulation results.

Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

For two matroids $$M_1$$ and $$M_2$$ with the same ground set V and two cost functions $$w_1$$ and $$w_2$$ on $$2^V$$ , we consider the problem of finding bases $$X_1$$ of $$M_1$$ and $$X_2$$ of $$M_2$$ minimizing $$w_1(X_1)+w_2(X_2)$$ subject to a certain cardinality constraint on their intersection $$X_1 \cap X_2$$ . For this problem, Lendl et al. (Matroid bases with cardinality constraints on the intersection, arXiv:1907.04741v2 , 2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $$|X_1 \cap X_2|\le k$$ or $$|X_1 \cap X_2|\ge k$$ ; and designed a new primal-dual algorithm for the case where the constraint is $$|X_1 \cap X_2|=k$$ . The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $$\mathrm {M}$$ -convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.

Logarithmic Regret for Distributed Online Subgradient Method over Unbalanced Directed Networks

This paper investigates a constrained distributed online optimization problem over strongly connected communication networks, where a local cost function of each agent varies in time due to environmental factors. We propose a distributed online projected subgradient method over unbalanced directed networks. The performance of the proposed method is evaluated by a regret which is defined by the error between the cumulative cost over time and the cost of the optimal strategy in hindsight. We show that a logarithmic regret bound can be achieved for strongly convex cost functions. We also demonstrate the validity of the proposed method through a numerical example on distributed estimation over a diffusion field.

Fixed-Time Projection Algorithm for Distributed Constrained Optimization on Time-Varying Digraphs

This article studies the distributed convex optimization problem with a common decision variable, a global inequality constraint, and local constraint sets over a time-varying multiagent network, the objective function of which is a sum of agents’ local convex cost functions. To solve such problem, a penalty-based distributed continuous-time subgradient algorithm with time-varying gain is developed for each agent to seek the saddle point of the penalty Lagrangian function. It is shown that an exact primal optimal solution can be obtained with certain assumption on time-varying gain. Moreover, the proposed algorithm adopts fixed-time projection scheme to ensure that for any initial state value, each local state estimate converges to its convex constraint set within fixed time. Finally, numerical examples are provided to show the effectiveness of the theoretical results.

On the Accelerated Convergence of the Decentralized Event-triggered Algorithm for Convex Optimization

This article considers a problem of solving the optimal solution of the sum of locally convex cost functions over an undirected network. Each local convex cost function in the network is accessed only by each unit. To be able to reduce the amount of computation and get the desired result in an accelerated way, we put forward a fresh accelerated decentralized event-triggered algorithm, named as A-DETA, for the optimization problem. A-DETA combines gradient tracking and two momentum accelerated terms, adopts nonuniform step-sizes and emphasizes that each unit interacts with neighboring units independently only at the sampling time triggered by the event. On the premise of assuming the smoothness and strong convexity of the cost function, it is proved that A-DETA can obtain the exact optimal solution linearly in the event of sufficiently small positive step-size and momentum coefficient. Moreover, an explicit linear convergence speed is definitely shown. Finally, extensive simulation example validates the usability of A-DETA.

A Combination Interval Prediction Model Based on Biased Convex Cost Function and Auto-Encoder in Solar Power Prediction

Due to the intermittent and stochastic nature of solar power, solar power interval prediction is of great importance for grid management and power dispatching. A combination interval prediction model based on the lower and upper bound estimation (LUBE) is proposed to efficiently quantify the solar power prediction uncertainty. In the proposed model, the upper and lower bounds are separately predicted by two prediction engines. The extreme learning machine (ELM) is selected as the basic prediction engine. The auto-encoder technique is used to initialize the input weight matrix of ELM for efficient feature learning. A novel biased convex cost function is developed for ELM to predict the interval boundary. The output weight matrix of ELM can be solved via the convex optimization technique instead of the conventional heuristic algorithm. The proposed interval prediction model can be formulated as a bi-level optimization problem. In the lower-level problem, the lower and upper ELMs are trained under different candidate hyper-parameters of the biased cost function. In the upper-level problem, the optimal combination of the lower and upper prediction engines is determined by evaluating the interval prediction performance. Comprehensive experiments based on public data set are conducted to validate the superiority of the proposed interval prediction model.

Stochastic optimal transport revisited

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker-Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the mather theory. The superposition principle is the construction of a semimartingale from the Fokker-Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by E. Nelson, called Nelson's problem, and was proved by E. Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.

A New Combinatorial Algorithm for Separable Convex Resource Allocation with Nested Bound Constraints

The separable convex resource allocation problem with nested bound constraints aims to allocate B units of resources to n activities to minimize a separable convex cost function, with lower and upper bounds on the total amount of resources that can be consumed by nested subsets of activities. We develop a new combinatorial algorithm to solve this model exactly. Our algorithm is capable of solving instances with millions of activities in several minutes. The running time of our algorithm is at most 73% of the running time of the current best algorithm for benchmark instances with three classes of convex objectives. The efficiency of our algorithm derives from a combination of constraint relaxation and divide and conquer based on infeasibility information. In particular, nested bound constraints are relaxed first; if the solution obtained violates some bound constraints, we show that the problem can be divided into two subproblems of the same structure and smaller sizes according to the bound constraint with the largest violation. Summary of Contribution. The resource allocation problem is a collection of optimization models with a wide range of applications in production planning, logistics, portfolio management, telecommunications, statistical surveys, and machine learning. This paper studies the resource allocation model with prescribed lower and upper bounds on the total amount of resources consumed by nested subsets of activities. These nested bound constraints are motivated by storage limits, time-window requirements, and budget constraints in various applications. The model also appears as a subproblem in models for green logistics and machine learning, and it has to be solved repeatedly. The model belongs to the class of computationally challenging convex mixed-integer nonlinear programs. We develop a combinatorial algorithm to solve this model exactly. Our algorithm is faster than the algorithm that currently has the best theoretical complexity in the literature on an extensive set of test instances. The efficiency of our algorithm derives from the combination of an infeasibility-guided divide-and-conquer framework and a scaling-based greedy subroutine for resource allocation with submodular constraints. This paper also showcases the prevalent mismatch between the theoretical worst-case time complexity of an algorithm and its practical efficiency. We have offered some explanations of this mismatch through the perspectives of worst-case analysis, specially designed instances, and statistical metrics of numerical experiments. The implementation of our algorithm is available on an online repository.

Functional penalised basis pursuit on spheres

In this paper, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere. More specifically, we consider recovering spherical fields directly in the continuous domain using functional penalised basis pursuit problems with gTV regularisation terms. Our framework is compatible with various measurement types as well as non-differentiable convex cost functionals. Via a novel representer theorem, we characterise their solution sets in terms of spherical splines with sparse innovations. We use this result to derive an approximate canonical spline-based discretisation scheme, with vanishing approximation error. To solve the resulting finite-dimensional optimisation problem, we propose an efficient and provably convergent primal-dual splitting algorithm. We illustrate the versatility of our framework on real-life examples from the field of environmental sciences.

Real-Time Distributed Economic Dispatch Adapted to General Convex Cost Functions: A Secant Approximation-Based Method

The increasing penetration of distributed energy resources (DERs) in power systems has aroused interest in distributed economic dispatch (DED). While quadratic cost functions are usually adopted, those of DERs are not necessarily quadratic in practice. Existing distributed algorithms perform unsatisfying convergence rates under nonquadratic cost functions. Therefore, this article proposes a secant approximation-based method (SAM) for general convex cost functions to achieve efficient convergence. The marginal cost functions are represented by linear functions in particular price intervals. With a dynamic ratio consensus (DRC) approach, the DERs with linear marginal cost functions can rapidly reach price consensus. The approximated prices can, in turn, determine new price intervals, which are narrower and more accurate.Case studies based on real-world data verify the improved convergence rate under general convex cost functions and exhibit growing performance in larger-scale cases. The proposed algorithm can be applied to DED in systems with high penetration of DERs and massive agents with general convex cost functions.

On Weak Super Ricci Flow through Neckpinch

Abstract In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊 n . We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.

The impact of congestion on protection decisions in supply networks under disruptions

We analyze the impact of congestion on the protection decisions of supply networks for mitigating major disruption. We present a tri-level mixed integer programming model to identify critical facilities to secure and backup plans with appropriate capacity levels and response speeds during disruption. The congestion arising at the facilities as a result of flow reallocation from the disrupted facilities is modelled as a convex cost function. We present an implicit enumeration algorithm to solve the tri-level model. The computational results demonstrate the efficiency of the solution method. Our analysis depicts that a decentralized protection strategy is recommended under congestion.

An Approximate Distributed Gradient Estimation Method for Networked System Optimization Under Limited Communications

This paper considers the networked system optimization problem by cooperatively finding an approximately optimal solution to the overall network convex cost function, which is the sum of the individual cost functions of nodes (agents) in a networked system. A new distributed gradient descent algorithm is proposed based on the distributed estimation of the sum of gradients of individual cost functions using a consensus-type coordination algorithm. The proposed algorithm can address unknown directed communication topologies and only requires limited communications and information exchanges among nodes in the networked system. Under the assumption that the communication graph is strongly connected, the convergence of the proposed algorithm is rigorously analyzed. Simulation examples are presented to demonstrate the applicability and effectiveness of the proposed algorithm.

Robustification of Continuous-Time ADMM against Communication Delays under Non-Strict Convexity: A Passivity-Based Approach

In this paper, we address a class of distributed optimization problems with non-strictly convex cost functions in the presence of communication delays between an agent and a coordinator. To this en...