We introduce two optimization methods for Difference-of-Convex (DC) optimization problems involving hard and soft constraints. While hard-constraints are strict requirements that must be satisfied for a solution to be valid, soft constraints are preferences that are desirable but not mandatory to be met. In this work, the hard constraints are considered convex, while the soft constraints (potentially nonconvex) are incorporated into the objective function via squared distance-to-set penalty terms. Our first algorithm requires only a difference of convex and weakly convex (CwC) decomposition of the objective function – a milder assumption than the standard DC decomposition, while preserving implementability and broadening the method's applicability. The second algorithm is an original bundle method that leverages a novel self-stabilizing model and operates under a standard DC decomposition framework. The proposed implementable algorithms come with convergence guarantees to critical points of the penalized, nonsmooth, nonconvex, optimization model. The theoretical framework is grounded in variational analysis and nonsmooth optimization, and our approaches have potential applications in signal processing, machine learning, and operations research.

An overview and comparison of spectral bundle methods for primal and dual semidefinite programs

A novel algorithm was proposed for solving the max-cut problem, which seeks to identify the cut with the maximum weight in a given graph. Our technique is based on the bundle approach, applied to a newly formulated semidefinite relaxation. This research establishes the theoretical convergence of our approximation technique and presents the numerical results obtained on several large-scale graphs from the BiqMac library, specifically with 100, 250, and 500 nodes. The resulting performance was compared with that produced by two alternative semidefinite programming-based approximation methods, namely the BiqMac and BiqBin solvers, by comparing the CPU time and the number of function calls. The primary objective of this work was to enhance the scalability and computational efficiency in solving the max-cut problem, particularly for large-scale graph instances. Despite the development of numerous approximation algorithms, a persistent challenge lies in effectively handling problems with a large number of constraints. Our algorithm addresses this by integrating a novel semidefinite relaxation with a bundle-based optimization framework, achieving faster convergence and fewer function calls. These advancements mark a meaningful step forward in the efficient resolution of NP-hard combinatorial optimization problems.

In this paper, we present an approximating scheme based on the proximal point algorithm for solving generalized fractional programs involving ratios of difference of convex (DC) functions and subject to DC constraints, which we shall refer to as DC-GFP. These problems are usually nonsmooth and nonconvex, but we approximate them iteratively with parametric convex ones. We capitalize on the latter attribute to employ the conventional bundle method to address them. The proposed method is seen as a pure proximal algorithm or a proximal bundle method and generates a sequence of approximate solutions that converge to critical points satisfying the necessary optimality conditions of the KKT type.

We develop an algorithm based on the idea of the bundle trust-region method to solve nonsmooth nonconvex constrained optimization problems. The resulting algorithm inherits some attractive features from both bundle and trust-region methods. Moreover, it allows effective control of the size of trust-region subproblems via the compression and aggregation techniques of bundle methods. On the other hand, the trust-region strategy is used to manage the search region and accept a candidate point as a new successful iterate. Global convergence of the developed algorithm is studied under some mild assumptions and its encouraging preliminary computational results are reported.

Given the increasing global emphasis on sustainable energy usage and the rising energy demands of cellular wireless networks, this work seeks an optimal short-term, continuous-time power-procurement schedule to minimize operating expenditure and the carbon footprint of cellular wireless networks equipped with energy-storage capacity, and hybrid energy systems comprising uncertain renewable energy sources. Despite the stochastic nature of wireless fading channels, the network operator must ensure a certain quality-of-service (QoS) constraint with high probability. This probabilistic constraint prevents using the dynamic programming principle to solve the stochastic optimal control problem. This work introduces a novel time-continuous Lagrangian relaxation approach tailored for real-time, near-optimal energy procurement in cellular networks, overcoming tractability problems associated with the probabilistic QoS constraint. The numerical solution procedure includes an efficient upwind finite-difference solver for the Hamilton-Jacobi-Bellman equation corresponding to the relaxed problem, and an effective combination of the limited memory bundle method (LMBM) for handling nonsmooth optimization and the stochastic subgradient method (SSM) to navigate the stochasticity of the dual problem. Numerical results, based on the German power system and daily cellular traffic data, demonstrate the computational efficiency of the proposed numerical approach, providing a near-optimal policy in a practical timeframe.

Abstract We propose the novel p‐branch‐and‐bound method for solving two‐stage stochastic programming problems whose deterministic equivalents are represented by non‐convex mixed‐integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p‐branch‐and‐bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p‐Lagrangian decomposition, generates a mixed‐integer relaxation of a dual problem with a separable structure for a primal non‐convex MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and non‐anticipativity conditions are met once the optimal solution is obtained. This paper also presents a comparative analysis of the p‐branch‐and‐bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank–Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank–Wolfe method as an inner loop in the classic progressive hedging. The p‐branch‐and‐bound method's efficiency was tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi.

Numerical simulation study on the impact of convective heat transfer on lithium battery air cooling thermal model

Entry trajectory optimization of lifting-body vehicle by successive difference-of-convex programming

Semi-infinite minimax problems are widely utilized in various fields; however, there is a scarcity of algorithms that can directly tackle convex-convex and convex-concave semi-infinite minimax problems. An inexact algorithm based on the bundle method is introduced in this paper, which can be directly applied to solve both types of semi-infinite minimax problems. The novel algorithm offers the advantage of not requiring exact solutions for the inner maximization problem but only necessitates optimal solution with a certain level of precision. Additionally, the augmentation function method is employed to address nonconvergence issues encountered in traditional bundle method when dealing with convex-convex minimax problems. Global convergence of our algorithm is proven under reasonable assumptions. Numerical results from several examples demonstrate the effectiveness and practicality of our proposed approach.

During the past decade, many diagnostic instruments have been developed that utilize electronic pulse dilation to achieve temporal resolution in the sub-10 ps range. The motivation behind these development efforts was the need for advanced diagnostics in high-density physics experiments around the world. This technology converts the signal of interest into a free electron cloud, which is accelerated into a vacuum drift space. The acceleration potential varies over time and causes axial velocity dispersion in the electron cloud. This velocity dispersion is converted into time separation after electrons pass through drift space. Then, traditional time resolved methods were used to detect free electrons, and the effective temporal resolution was magnified many times. A gated microchannel plate (MCP) X-ray framing camera based on pulse-dilation technology has been designed and manufactured in the paper. Here, we discuss design details and applications of these instruments. The temporal resolution measured without using broadening technology is approximately 78 ps. When the excitation pulse is applied to the PC, the pulse dilation technique is used to increase the measured temporal resolution to 9 ps. The propagation speed of gated pulses in MCP microstrip lines was measured using fiber bundle method, which is approximately 1.8 × 108 m/s.

Abstract This work considers a short‐term, continuous time setting characterized by a coupled power supply system controlled exclusively by a single provider and comprising a cascade of hydropower systems (dams), fossil fuel power stations, and a storage capacity modeled by a single large battery. Cascaded hydropower generators introduce time‐delay effects in the state dynamics, which are modeled with differential equations, making it impossible to use classical dynamic programming. We address this issue by introducing a novel Lagrangian relaxation technique over continuous‐time constraints, constructing a nearly optimal policy efficiently. This approach yields a convex, nonsmooth optimization dual problem to recover the optimal Lagrangian multipliers, which is numerically solved using a limited memory bundle method. At each step of the dual optimization, we need to solve an optimization subproblem. Given the current values of the Lagrangian multipliers, the time delays are no longer active, and we can solve a corresponding nonlinear Hamilton–Jacobi–Bellman (HJB) Partial Differential Equation (PDE) for the optimization subproblem. The HJB PDE solver provides both the current value of the dual function and its subgradient, and is trivially parallelizable over the state space for each time step. To handle the infinite‐dimensional nature of the Lagrange multipliers, we design an adaptive refinement strategy to control the duality gap. Furthermore, we use a penalization technique for the constructed admissible primal solution to smooth the controls while achieving a sufficiently small duality gap. Numerical results based on the Uruguayan power system demonstrate the efficiency of the proposed mathematical models and numerical approach.

We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle for computing function values and subgradients. The algorithm allows for an arbitrary number of master problem processes computing new candidate points and oracle processes evaluating functions at those candidate points. These processes share information by communication with a single supervisor process that resembles the main loop of a classic bundle method. All processes run in parallel and no explicit synchronization step is required. Instead, the asynchronous and possibly outdated results of the oracle computations can be seen as an inexact function oracle. Hence, we show the convergence of our method under weak assumptions very similar to inexact and incremental bundle methods. In particular, we show how the algorithm learns important structural properties of the functions to control the inaccuracy induced by the asynchronicity automatically such that overall convergence can be guaranteed.

Based on the theory of zero bending moment under constant load, various optimization methods exist for the top beam of large-span continuous rigid frame bridges. These include achieving zero bending moment at the root of the cantilever beam, at the control stage section, and through the zero deflection method. This study aims to explore the methods and effects of optimizing roof beam design using the constant load zero bending moment method and the “three group bundle method”. Using finite element modeling, the total number and eccentricity of prestressed tendons required for each suspended pouring block are determined. Additionally, the “three group beam matching method” is employed to adjust the steel beam, adhering to the design concept of “large cantilever beam matching and small cantilever beam matching”, to achieve a reasonable configuration of the top plate beam. Through specific engineering examples, the results demonstrate that utilizing the constant load zero moment method and the “three group bundle method” can significantly enhance the structural performance and economy of large-span continuous rigid frame bridges. Moreover, it offers practical operability, providing an important reference basis for similar project designs.

Proximal bundle methods are a class of optimisation algorithms that leverage the proximal operator to address nonsmoothness in the objective function efficiently. This study focuses on a derivative-free (DFO) proximal bundle method and one of its applications called the DFO VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-algorithm. These algorithms incorporate approximate proximal points as subprocedures in order to optimise convex nonsmooth functions based on approximated subdifferential information. Interestingly, the classical VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-algorithm, which operates on true subgradient values, achieves superlinear convergence. At each iteration, the algorithm divides the whole space into two: the smooth U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{U}$\\end{document}-space and the nonsmooth V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{V}$\\end{document}-space. It takes a Newton-like step on the U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{U}$\\end{document}-space and a proximal-point step on the V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{V}$\\end{document}-space, enabling it to handle both smooth and nonsmooth parts effectively and converge faster. In this work, we reveal the worst possible convergence rate for the DFO VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-method by showing the linear convergence of the DFO proximal bundle method. This will be done by presenting a suitable framework and using the subdifferential-based error bound on the distance to critical points.

ABSTRACT We propose a descent subgradient algorithm for minimizing a function f : R n → R , assumed to be locally Lipschitz, but not necessarily smooth or convex. To find an effective descent direction, the Goldstein ε-subdifferential is approximated through an iterative process. The method enjoys a new two-point variant of Mifflin's line search in which the subgradients are arbitrary. Thus, the line search procedure is easy to implement. Moreover, in comparison to bundle methods, the quadratic subproblems have a simple structure, and to handle nonconvexity the proposed method requires no algorithmic modification. We study the global convergence of the method and prove that any accumulation point of the generated sequence is Clarke stationary, assuming that the objective f is weakly upper semismooth. We illustrate the efficiency and effectiveness of the proposed algorithm on a collection of academic and semi-academic test problems.

Algorithms for solving certain nonconvex, nonsmooth, finite-sum optimization problems are proposed and tested. In particular, the algorithms are proposed and tested in the context of a transductive support vector machine (TSVM) problem formulation arising in semi-supervised machine learning. The common feature of all algorithms is that they employ an incremental quasi-Newton (IQN) strategy, specifically an incremental BFGS (IBFGS) strategy. One applies an IBFGS strategy to the problem directly, whereas the others apply an IBFGS strategy to a difference-of-convex reformulation, smoothed approximation, or (strongly) convex local approximation. Experiments show that all IBFGS approaches fare well in practice, and all outperform a state-of-the-art bundle method when solving TSVM problem instances.

We consider optimization problems with objective and constraint being the difference of convex and weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimization problems, particularly those involving certain classes of composite and nonconvex value functions. We investigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et al., Comput. Optim. Appl., 78 (2021), pp. 451-490 ] to compute critical points for problems in this class. Our modifications on that algorithm boil down to a different master program and an original rule to update the proximal parameter to ensure convergence in this more general setting. Thanks to this new rule, no pre-estimation of the underlying weakly-convex moduli is needed, opening the way to deal with optimization problems for which no practical and mathematically sound algorithms exist. Numerical experiments on some nonconvex stochastic problems illustrate the practical performance of the method.

We introduce a bundle method for the unconstrained minimization of nonsmooth difference-of-convex (DC) functions, and it is based on the calculation of a special type of descent direction called descent–ascent direction. The algorithm only requires evaluations of the minuend component function at each iterate, and it can be considered as a parsimonious bundle method as accumulation of information takes place only in case the descent–ascent direction does not provide a sufficient decrease. No line search is performed, and proximity control is pursued independent of whether the decrease in the objective function is achieved. Termination of the algorithm at a point satisfying a weak criticality condition is proved, and numerical results on a set of benchmark DC problems are reported. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms – Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0142 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0142 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

This repository contains the code to run the experiments present in "The descent-ascent algorithm for DC programming", a bundle method for the unconstrained minimization of nonsmooth difference-of-convex functions.