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- Relaxations and cutting planes for linear programs with complementarity
constraints-
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Abstract: Abstract We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and contains the extended formulation obtained from the ERLT introduced by Nguyen, Richard, and Tawarmalani as a special case. We demonstrate how to obtain strong cutting planes for our formulation from both the stable set polytope and the boolean quadric polytope associated with a complete bipartite graph. Through an extensive computational study for three types of practical problems, we assess the performance of our proposed linear relaxation and new cutting-planes in terms of the optimality gap closed.
PubDate: 2024-09-01
- An approximation proximal gradient algorithm for nonconvex-linear minimax
problems with nonconvex nonsmooth terms-
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Abstract: Abstract Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an \(\varepsilon \) -stationary point is upper bounded by \({\mathcal {O}}(\varepsilon ^{-3})\) . Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm.
PubDate: 2024-09-01
- Optimality conditions and sensitivity analysis in parametric nonconvex
minimax programming-
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Abstract: Abstract In this paper, we perform optimality conditions and sensitivity analysis for parametric nonconvex minimax programming problems. Our aim is to study the necessary optimality conditions by using the Mordukhovich (limiting) subdifferential and to give upper estimations for the Mordukhovich subdifferential of the optimal value function in the problem under consideration. The optimality conditions and sensitivity analysis are obtained by using upper estimates for Mordukhovich subdifferentials of the maximum function. The results on optimality conditions are then applied to parametric multiobjective optimization problems. An example is given to illustrate our results.
PubDate: 2024-09-01
- Gradient projection method on the sphere, complementarity problems and
copositivity-
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Abstract: Abstract By using a constant step-size, the convergence analysis of the gradient projection method on the sphere is presented for a closed spherically convex set. This algorithm is applied to discuss copositivity of operators with respect to cones. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. To our best knowledge this is the first numerical method related to the copositivity of operators with respect to the positive semidefinite cone. Numerical results concerning the copositivity of operators are also provided.
PubDate: 2024-09-01
- Distributed accelerated gradient methods with restart under quadratic
growth condition-
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Abstract: Abstract We consider solving convex problems satisfying quadratic growth condition (QGC) over a distributed setting with no central server. Such problems are popular in distributed machine learning applications. When QGC growth parameter c is known, we propose distributed accelerated gradient methods with restarts, named PDACA and DACA respectively for constrained and unconstrained settings. In practical problems when c is unavailable, we design mPDACA and mDACA methods respectively for constrained and unconstrained settings, where novel distributed mechanisms are proposed to update the estimates of growth parameter c using only local quantities depending on local proximal operators or local gradients. We further derive theoretical guarantees and gradient computation and communication complexities for all four proposed algorithms. Extensive numerical experiments on logistic regression on different communication topologies showcase the utility of our algorithms in comparison with baseline methods.
PubDate: 2024-09-01
- A new optimization approach to solving split equality problems in Hilbert
spaces-
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Abstract: Abstract We introduce a new optimization approach to solving systems of split equality problems in real Hilbert spaces. We use the inertial method in order to improve the convergence rate of the proposed algorithms. Our algorithms do not depend on the norms of the bounded linear operators which appear in each split equality problem of the system under consideration. This is also a strong point of our algorithms because it is known that it is difficult to compute or estimate the norm of a linear operator in the general case.
PubDate: 2024-09-01
- Almost optimal manipulation of pairwise comparisons of alternatives
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Abstract: Abstract The role of an expert in the decision-making process is crucial. If we ask an expert to help us to make a decision we assume their honesty. But what if the expert is dishonest' Then, the answer on how difficult it is for an expert to provide manipulated data in a given case of decision-making process becomes essential. In the presented work, we consider manipulation of a ranking obtained by the Geometric Mean Method applied to a pairwise comparisons matrix. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives in a ranking. We also define a new index which measures how difficult such manipulation is in a given case.
PubDate: 2024-09-01
- Social equity in international environmental agreements
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Abstract: Abstract The aim of this paper is to investigate the problem of designing and building International Environmental Agreements (IEAs) taking into account some normative properties. We consider n asymmetric countries of the world, each one generating a quantity of pollutant emissions from the production of goods and services. We assume that individual emissions yield private benefits and negative externalities affecting all countries. To determine its own level of pollution, each state conducts a cost-benefit analysis. The absence of a supranational entity imposing emissions reduction makes IEAs based on voluntary participation. Examining the standard static non-cooperative game-theoretical model of coalition formation, we discover that the resulting emissions allocations might not be equitable à la Foley. It means that there might exist at least one player preferring to implement some other agent’s strategic plan instead of to play her own strategy. With the goal of studying whether equity, at least among coalesced countries, may be a criterion leading to social improvement, we introduce a new optimization rule. We require that members of an environmental coalition have to solve the maximization problem subject to the constraint imposing that they do not envy each other. Analyzing the particular case of two-player games, we get that, when countries are, in a sense, not too different from each other, our new mechanism endogenously induces social equity. By imposing a suitable total emission cap, the same results extend to all those games where our and standard solutions coexist and are different.
PubDate: 2024-09-01
- Subspace Newton method for sparse group $$\ell _0$$ optimization problem
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Abstract: Abstract This paper investigates sparse optimization problems characterized by a sparse group structure, where element- and group-level sparsity are jointly taken into account. This particular optimization model has exhibited notable efficacy in tasks such as feature selection, parameter estimation, and the advancement of model interpretability. Central to our study is the scrutiny of the \(\ell _0\) and \(\ell _{2,0}\) norm regularization model, which, in comparison to alternative surrogate formulations, presents formidable computational challenges. We embark on our study by conducting the analysis of the optimality conditions of the sparse group optimization problem, leveraging the notion of a \(\gamma \) -stationary point, whose linkage to local and global minimizer is established. In a subsequent facet of our study, we develop a novel subspace Newton algorithm for sparse group \(\ell _0\) optimization problem and prove its global convergence property as well as local second-order convergence rate. Experimental results reveal the superlative performance of our algorithm in terms of both precision and computational expediency, thereby outperforming several state-of-the-art solvers.
PubDate: 2024-09-01
- Characteristic sets and characteristic numbers of matrix two-person games
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Abstract: Abstract Focusing on the extreme points of the solution sets of matrix two-person games, we propose the notions of characteristic sets and characteristic numbers. The characteristic sets (resp., the characteristic numbers) are the sets (resp., the numbers) of the extreme points of the solution set of the game and the optimal solution sets of the players. These concepts allow us to measure the complexity of the game. The larger the characteristic numbers, the more complex the game is. Among other things, we obtain upper bounds for the characteristic numbers and give a novel geometric construction. By the construction, we get useful descriptions of the optimal strategy set of each player of a game given by any nonsingular square matrix. Namely, the investigation of the geometry the just-mentioned sets reduces to computing or studying certain simpler sets. We also formulate several open problems.
PubDate: 2024-09-01
- Budget-constrained profit maximization without non-negative objective
assumption in social networks-
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Abstract: Abstract In this paper, we study the budget-constrained profit maximization problem with expensive seed endorsement, a derivation of the well-studied influence maximization and profit maximization in social networks. While existing research requires the non-negativity of the objective profit function, this paper considers real-world scenarios where costs may surpass revenue. Specifically, our problem can be regarded as maximizing the difference between a non-negative submodular function and a non-negative modular function under a knapsack constraint, allowing for negative differences. To tackle this challenge, we propose two algorithms. Firstly, we employ a twin greedy and enumeration technique to design a polynomial-time algorithm with a quarter weak approximation ratio, providing a balance between computational efficiency and solution quality. Then, we incorporate a threshold decreasing technique to enhance the time complexity of the first algorithm, yielding an improved computational efficiency while maintaining a reasonable level of solution accuracy. To our knowledge, this is the first paper to study the profit maximization beyond non-negativity and to propose polynomial-time algorithms with a constant bicriteria approximation ratio.
PubDate: 2024-08-14
- A two-phase sequential algorithm for global optimization of the standard
quadratic programming problem-
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Abstract: Abstract We introduce a new sequential algorithm for the Standard Quadratic Programming Problem (StQP), which exploits a formulation of StQP as a Linear Program with Linear Complementarity Constraints (LPLCC). The algorithm is finite and guarantees at least in theory a \(\delta \) -approximate global minimum for an arbitrary small \(\delta \) , which is a global minimum in practice. The sequential algorithm has two phases. In Phase 1, Stationary Points (SP) with strictly decreasing objective function values are computed. Phase 2 is designed for giving a certificate of global optimality for the last SP computed in Phase 1. Two different Nonlinear Programming Formulations for LPLCC are proposed for each one of these phases, which are solved by efficient enumerative algorithms. New procedures for computing a lower bound for StQP are also proposed, which are easy to implement and give tight bounds in general. Computational experiments with a number of test problems from known sources indicate that the two-phase sequential algorithm is, in general, efficient in practice. Furthermore, the algorithm seems to be an efficient way to study the copositivity of a matrix by exploiting an StQP with this matrix.
PubDate: 2024-08-12
- The limitation of neural nets for approximation and optimization
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Abstract: Abstract We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function for approximating the objective functions of popular nonlinear optimization test problems, and the evidence provided shows that ReLU and SiLU exhibit the best performance on both training and testing data. We then analyze the accuracy of function value, gradient, and Hessian approximations for such objective functions obtained through interpolation/regression models and neural networks. When compared to interpolation/regression models, neural networks can deliver competitive zero- and first-order approximations (at a high training cost) but underperform on second-order approximation. However, it is shown that combining a neural net activation function with the natural basis for quadratic interpolation/regression can waive the necessity of including cross terms in the natural basis, leading to models with fewer parameters to determine. Lastly, we provide evidence that the performance of a state-of-the-art derivative-free optimization algorithm can hardly be improved when the gradient of an objective function is approximated using any of the surrogate models considered, including neural networks.
PubDate: 2024-08-09
- Relaxed projection methods for solving variational inequality problems
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Abstract: Abstract In this paper, we introduce a new relaxed projection approach for solving the variational inequality problems in a real Hilbert space. First, we propose a solution mapping and show its strongly quasi-nonexpansive properties. Next, we apply the mapping to present two algorithms for solving partially pseudomonotone variational inequality problems and split variational inequality problems. Weak convergence of the algorithms is showed under partially pseudomonotone and Lipschitz continuous assumptions of the cost mappings. Finally, we give some numerical results for the proposed algorithms and comparison with other known methods.
PubDate: 2024-08-08
- Novel algorithms based on forward-backward splitting technique: effective
methods for regression and classification-
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Abstract: Abstract In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios.
PubDate: 2024-08-07
- Computing the recession cone of a convex upper image via convex projection
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Abstract: Abstract It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.
PubDate: 2024-08-01
- A nonmonotone accelerated proximal gradient method with variable stepsize
strategy for nonsmooth and nonconvex minimization problems-
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Abstract: Abstract In this paper, we consider the problem that minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting, which arising in many contemporary applications such as machine learning, statistics, and signal/image processing. To solve this problem, we propose a new nonmonotone accelerated proximal gradient method with variable stepsize strategy. Note that incorporating inertial term into proximal gradient method is a simple and efficient acceleration technique, while the descent property of the proximal gradient algorithm will lost. In our algorithm, the iterates generated by inertial proximal gradient scheme are accepted when the objective function values decrease or increase appropriately; otherwise, the iteration point is generated by proximal gradient scheme, which makes the function values on a subset of iterates are decreasing. We also introduce a variable stepsize strategy, which does not need a line search or does not need to know the Lipschitz constant and makes the algorithm easy to implement. We show that the sequence of iterates generated by the algorithm converges to a critical point of the objective function. Further, under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz inequality, we prove the convergence rates of the objective function values and the iterates. Moreover, numerical results on both convex and nonconvex problems are reported to demonstrate the effectiveness and superiority of the proposed method and stepsize strategy.
PubDate: 2024-08-01
- Nonlinear scalarization in set optimization based on the concept of null
set-
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Abstract: Abstract The aim of this paper is to introduce a nonlinear scalarization function in set optimization based on the concept of null set which was introduced by Wu (J Math Anal Appl 472(2):1741–1761, 2019). We introduce a notion of pseudo algebraic interior of a set and define a weak set order relation using the concept of null set. We investigate several properties of this nonlinear scalarization function. Further, we characterize the set order relations and investigate optimality conditions for solution sets in set optimization based on the concept of null set. Finally, a numerical example is provided to compute a weak minimal solution using this nonlinear scalarization function.
PubDate: 2024-08-01
- Sketch-based multiplicative updating algorithms for symmetric nonnegative
tensor factorizations with applications to face image clustering-
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Abstract: Abstract Nonnegative tensor factorizations (NTF) have applications in statistics, computer vision, exploratory multi-way data analysis, and blind source separation. This paper studies randomized multiplicative updating algorithms for symmetric NTF via random projections and random samplings. For random projections, we consider two methods to generate the random matrix and analyze the computational complexity, while for random samplings the uniform sampling strategy and its variants are examined. The mixing of these two strategies is then considered. Some theoretical results are presented based on the bounds of the singular values of sub-Gaussian matrices and the fact that randomly sampling rows from an orthogonal matrix results in a well-conditioned matrix. These algorithms are easy to implement, and their efficiency is verified via test tensors from both synthetic and real datasets, such as for clustering facial images.
PubDate: 2024-08-01
- Faster algorithms for sparse ILP and hypergraph multi-packing/multi-cover
problems-
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Abstract: Abstract In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in \({{\,\mathrm{\mathcal {P}}\,}}\cap {{\,\mathrm{\mathbb {Z}}\,}}^n\) , assuming that \({{\,\mathrm{\mathcal {P}}\,}}\) is a polyhedron, defined by systems \(A x \le b\) or \(Ax = b,\, x \ge 0\) with a sparse matrix A. We develop algorithms for these problems that outperform state-of-the-art ILP and counting algorithms on sparse instances with bounded elements in terms of the computational complexity. Assuming that the matrix A has bounded elements, our complexity bounds have the form \(s^{O(n)}\) , where s is the minimum between numbers of non-zeroes in columns and rows of A, respectively. For \(s = o\bigl (\log n \bigr )\) , this bound outperforms the state-of-the-art ILP feasibility complexity bound \((\log n)^{O(n)}\) , due to Reis & Rothvoss (in: 2023 IEEE 64th Annual symposium on foundations of computer science (FOCS), IEEE, pp. 974–988). For \(s = \phi ^{o(\log n)}\) , where \(\phi \) denotes the input bit-encoding length, it outperforms the state-of-the-art ILP counting complexity bound \(\phi ^{O(n \log n)}\) , due to Barvinok et al. (in: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp. 566–572, https://doi.org/10.1109/SFCS.1993.366830, 1993), Dyer, Kannan (Math Oper Res 22(3):545–549, https://doi.org/10.1287/moor.22.3.545, 1997), Barvinok, Pommersheim (Algebr Combin 38:91–147, 1999), Barvinok (in: European Mathematical Society, ETH-Zentrum, Zurich, 2008). We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximum Matching problems.
PubDate: 2024-08-01
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