arXiv subjects

Ellen H. Fukuda

Publications and source records attributed to Ellen H. Fukuda.

At least 19 recordsLinked to original sources

arXiv2026

A Regularized Nikaido-Isoda Function Approach to Multi-Leader-Follower Games

A multi-leader--follower game (MLFG) is a hierarchical noncooperative game in which leaders compete at the upper level while taking into account the followers' best responses at the lower level. A typical approach to solving the MLFG reformulates it as an equilibrium problem with equilibrium constraints (EPECs) by replacing the lower-level game with its KKT conditions. Another approach, when each follower's response is unique, is to reformulate the MLFG as a Nash equilibrium problem by substituting these response functions into each leader's problem. However, both reformulations may lack scalability since higher-order derivatives may be required when solving the resulting problems. In this paper, we propose a new reformulation of the MLFG by exploiting a regularized Nikaido--Isoda function and approximating the MLFG by a single-level differentiable Nash equilibrium problem with a penalty parameter. The proposed reformulation neither requires derivative information on the followers' game nor assumes convexity of each follower's problem; hence, it can handle a broader class of MLFGs. Under global subanalyticity, we analyze the mathematical relationship between equilibria of the original MLFG and the proposed reformulation.

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arXiv2026

An uncertainty model for positive-valued parameters with application to robust optimization

Many practical optimization problems involve uncertain parameters that are strictly positive. However, the most common uncertainty sets used in robust optimization are the box and the ellipsoidal sets, which may include non-positive values when the level of uncertainty is large. This can lead to overly conservative solutions or make the corresponding robust counterpart infeasible. To overcome this, in this paper, we propose a new uncertainty-set model that not only preserves positivity but is also computationally tractable. The proposed set uses a particular convex function that measures the variation of uncertain parameters from their nominal values. We can also write the dual reformulation of the associated robust problem. For the theoretical results, we show several properties of the proposed model, including analytical bounds that guide the choice of the uncertainty level, as well as a probabilistic guarantee result. To check the validity of our proposal, we consider photovoltaic-battery operation planning problems and support vector machines in the numerical experiments. For these problems, standard uncertainty models may lead to infeasibility of the robust counterpart, while the proposed uncertainty set gives a tractable dual reformulation.

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arXiv2026

Biobjective optimization with M-convex functions

In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the special case where the M$^\natural$-convex function is M-convex. Additionally, we present a polynomial-time method for biobjective optimization problems that combine M$^\natural$-convex function minimization with lexicographic optimization.

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arXiv2025

Riemannian conditional gradient methods for composite optimization problems

In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a retraction-based convex function. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of $\mathcal{O}(1/k)$ for the adaptive and diminishing step sizes, where $k$ denotes the number of iterations. Additionally, we derive an iteration complexity of $\mathcal{O}(1/ε^2)$ for the Armijo step-size strategy to achieve $ε$-optimality, where $ε$ is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds.

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arXiv2025

Second-order sequential optimality conditions for nonlinear semidefinite optimization problems

Sequential optimality conditions play an important role in constrained optimization since they provide necessary conditions without requiring constraint qualifications (CQs). This paper introduces a second-order extension of the Approximate Karush-Kuhn-Tucker (AKKT) conditions, referred to as AKKT2, for nonlinear semidefinite optimization problems (NSDP). In particular, we provide a formal definition of AKKT2, as well as its stronger variant, called Complementary AKKT2 (CAKKT2), and prove that these conditions are necessary for local minima without any assumption. Moreover, under Robinson's CQ and the weak constant rank property, we show that AKKT2 implies the so-called weak second-order necessary condition. Finally, we propose a penalty-based algorithm that generates sequences whose accumulation points satisfy the AKKT2 and the CAKKT2 conditions.

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arXiv2025

An inertial iteratively regularized extragradient method for bilevel variational inequality problems

We study a bilevel variational inequality problem where the feasible set is itself the solution set of another variational inequality. Motivated by the difficulty of computing projections onto such sets, we consider a regularized extragradient method, as proposed by Samadi and Yousefian (2025), which operates over a simpler constraint set. Building on this framework, we introduce an inertial variant (called IneIREG) that incorporates momentum through extrapolation steps. We establish iteration-complexity bounds for the general (non-strongly monotone) case under both constant and diminishing regularization, and derive improved results under strong monotonicity assumptions. Our analysis extends and refines the results of the previous work by capturing both inertial and regularization effects within a unified framework. Preliminary numerical experiments are also presented to illustrate the behavior of the proposed method.

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arXiv2025

Optimality conditions for problems over symmetric cones and a simple augmented Lagrangian method

In this work we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second order cone programming and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show how to obtain second order optimality conditions for NSCPs in an easy manner, thus bypassing a number of difficulties associated to the usual variational analytical approach. We then discuss several aspects of this connection. In particular, we show a "sharp" criterion for membership in a symmetric cone that also encodes rank information. Also, we discuss the possibility of importing convergence results from nonlinear programming to NSCPs, which we illustrate by discussing a simple augmented Lagrangian method for nonlinear symmetric cones. We show that, employing the slack variable approach, we can use the results for NLPs to prove convergence results, thus extending a special case (i.e., the case with strict complementarity) of an earlier result by Sun, Sun and Zhang for nonlinear semidefinite programs.

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arXiv2025

A strong second-order sequential optimality condition for nonlinear programming problems

Most numerical methods developed for solving nonlinear programming problems are designed to find points that satisfy certain optimality conditions. While the Karush-Kuhn-Tucker conditions are well-known, they become invalid when constraint qualifications (CQ) are not met. Recent advances in sequential optimality conditions address this limitation in both first- and second-order cases, providing genuine optimality guarantees at local optima, even when CQs do not hold. However, some second-order sequential optimality conditions still require some restrictive conditions on constraints in the recent literature. In this paper, we propose a new strong second-order sequential optimality condition without CQs. We also show that a penalty-type method and an augmented Lagrangian method generate points satisfying these new optimality conditions.

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arXiv2025

A second-order sequential optimality condition for nonlinear second-order cone programming problems

In the last two decades, the sequential optimality conditions, which do not require constraint qualifications and allow improvement on the convergence assumptions of algorithms, had been considered in the literature. It includes the work by Andreani et al. (2017), with a sequential optimality condition for nonlinear programming, that uses the second-order information of the problem. More recently, Fukuda et al. (2023) analyzed the conditions that use second-order information, in particular for nonlinear second-order cone programming problems (SOCP). However, such optimality conditions were not defined explicitly. In this paper, we propose an explicit definition of approximate-Karush-Kuhn-Tucker 2 (AKKT2) and complementary-AKKT2 (CAKKT2) conditions for SOCPs. We prove that the proposed AKKT2/CAKKT2 conditions are satisfied at local optimal points of the SOCP without any constraint qualification. We also present two algorithms that are based on augmented Lagrangian and sequential quadratic programming methods and show their global convergence to points satisfying the proposed conditions.

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arXiv2024

A method for multi-leader-multi-follower games by smoothing the followers' response function

The multi-leader--multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader--multi-follower game (SLMFG). Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers' strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers' noncooperative game that is a function with leaders' strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers' response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments and confirm its validity.

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arXiv2023

Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport

In this paper, we propose nonlinear conjugate gradient methods for vector optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik conditions are extended for Riemannian manifolds. Specifically, we establish the existence of intervals of step sizes that satisfy the Wolfe conditions. The convergence analysis covers the vector extensions of the Fletcher--Reeves, conjugate descent, and Dai--Yuan parameters. Under some assumptions, we prove that the sequence obtained by the algorithm can converge to a Pareto stationary point. Moreover, we also discuss several other choices of the parameter. Numerical experiments illustrating the practical behavior of the methods are presented.

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arXiv2023

Monotonicity for Multiobjective Accelerated Proximal Gradient Methods

Accelerated proximal gradient methods, which are also called fast iterative shrinkage-thresholding algorithms (FISTA) are known to be efficient for many applications. Recently, Tanabe et al. proposed an extension of FISTA for multiobjective optimization problems. However, similarly to the single-objective minimization case, the objective functions values may increase in some iterations, and inexact computations of subproblems can also lead to divergence. Motivated by this, here we propose a variant of the FISTA for multiobjective optimization, that imposes some monotonicity of the objective functions values. In the single-objective case, we retrieve the so-called MFISTA, proposed by Beck and Teboulle. We also prove that our method has global convergence with rate $O(1/k^2)$, where $k$ is the number of iterations, and show some numerical advantages in requiring monotonicity.

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arXiv2023

An accelerated proximal gradient method for multiobjective optimization

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ($O(1 / k^2)$), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

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arXiv2023

New merit functions for multiobjective optimization and their properties

A merit (gap) function is a map that returns zero at the solutions of problems and strictly positive values otherwise. Its minimization is equivalent to the original problem by definition, and it can estimate the distance between a given point and the solution set. Ideally, this function should have some properties, including the ease of computation, continuity, differentiability, boundedness of the level set, and error boundedness. In this work, we propose new merit functions for multiobjective optimization with lower semicontinuous objectives, convex objectives, and composite objectives, and we show that they have such desirable properties under reasonable assumptions.

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arXiv2022

On the weak second-order optimality condition for nonlinear semidefinite and second-order cone programming

Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson's constraint qualification and a weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence results for first- and second-order algorithms. Since we are not assuming strict complementarity, the critical cone does not reduce to a subspace, thus, the second-order condition we arrive at is defined in terms of the lineality space of the critical cone. In the case of nonlinear programming, this condition reduces to the standard second-order condition widely used as second-order stationarity measure in the algorithmic practice.

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arXiv2022

A globally convergent fast iterative shrinkage-thresholding algorithm with a new momentum factor for single and multi-objective convex optimization

Convex-composite optimization, which minimizes an objective function represented by the sum of a differentiable function and a convex one, is widely used in machine learning and signal/image processing. Fast Iterative Shrinkage Thresholding Algorithm (FISTA) is a typical method for solving this problem and has a global convergence rate of $O(1 / k^2)$. Recently, this has been extended to multi-objective optimization, together with the proof of the $O(1 / k^2)$ global convergence rate. However, its momentum factor is classical, and the convergence of its iterates has not been proven. In this work, introducing some additional hyperparameters $(a, b)$, we propose another accelerated proximal gradient method with a general momentum factor, which is new even for the single-objective cases. We show that our proposed method also has a global convergence rate of $O(1/k^2)$ for any $(a,b)$, and further that the generated sequence of iterates converges to a weak Pareto solution when $a$ is positive, an essential property for the finite-time manifold identification. Moreover, we report numerical results with various $(a,b)$, showing that some of these choices give better results than the classical momentum factors.

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arXiv2022

Convergence rates analysis of a multiobjective proximal gradient method

Many descent algorithms for multiobjective optimization have been developed in the last two decades. Tanabe et al. (Comput Optim Appl 72(2):339--361, 2019) proposed a proximal gradient method for multiobjective optimization, which can solve multiobjective problems, whose objective function is the sum of a continuously differentiable function and a closed, proper, and convex one. Under reasonable assumptions, it is known that the accumulation points of the sequences generated by this method are Pareto stationary. However, the convergence rates were not established in that paper. Here, we show global convergence rates for the multiobjective proximal gradient method, matching what is known in scalar optimization. More specifically, by using merit functions to measure the complexity, we present the convergence rates for non-convex ($O(\sqrt{1 / k})$), convex ($O(1 / k)$), and strongly convex ($O(r^k)$ for some $r \in (0, 1)$) problems. We also extend the so-called Polyak-Łojasiewicz (PL) inequality for multiobjective optimization and establish the linear convergence rate for multiobjective problems that satisfy such inequalities ($O(r^k)$ for some $r \in (0, 1)$).

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arXiv2022

A revised sequential quadratic semidefinite programming method for nonlinear semidefinite optimization

In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently, complementarity-AKKT (CAKKT) conditions were also consider, as an alternative to the previous mentioned ones, that is more practical. Since few methods are shown to converge to CAKKT points, at least in conic optimization, and to complete the study associated to the SQSDP, here we propose a revised version of the method, maintaining the good properties. We modify the previous algorithm, prove the global convergence in the sense of CAKKT, and show some preliminary numerical experiments.

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