Wolfe Duality Research Articles

Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality

This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380]. 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.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

On directionally differentiable multiobjective programming problems with vanishing constraints

In this paper, a class of directionally differentiable multiobjective programming problems with inequality, equality and vanishing constraints is considered. Under both the Abadie constraint qualification and the modified Abadie constraint qualification, the Karush–Kuhn–Tucker type necessary optimality conditions are established for such nondifferentiable vector optimization problems by using the nonlinear version Gordan theorem of the alternative for convex functions. Further, the sufficient optimality conditions for such directionally differentiable multiobjective programming problems with vanishing constraints are proved under convexity hypotheses. Furthermore, vector Wolfe dual problem is defined for the considered directionally differentiable multiobjective programming problem vanishing constraints and several duality theorems are established also under appropriate convexity hypotheses.

Sensitivity Analysis of the Maximal Value Function with Applications in Nonconvex Minimax Programs

In this paper, we perform a sensitivity analysis for the maximal value function, which is the optimal value function for a parametric maximization problem. Our aim is to study various subdifferentials for the maximal value function. We obtain upper estimates of Fréchet, limiting, and horizon subdifferentials of the maximal value function by using some sensitivity analysis techniques sophisticatedly. The derived upper estimates depend only on the union of all solutions and not on its convex hull or only one solution from the solution set. Finally, we apply the derived results to develop some new necessary optimality conditions for nonconvex minimax problems. In the nonconvex-concave setting, our Wolfe duality approach compares favorably with the first-order approach in that the necessary condition is sharper and the constraint qualification is weaker. Funding: L. Guo was supported by the National Natural Science Foundation of China [Grants 72131007, 72140006, and 12271161] and the Natural Science Foundation of Shanghai [Grant 22ZR1415900]. J. J. Ye was partially supported by the Natural Sciences and Engineering Research Council of Canada. J. Zhang was supported by the National Natural Science Foundation of China [Grant 12222106], the Shenzhen Science and Technology Program [Grant RCYX20200714114700072], and the Guangdong Basic and Applied Basic Research Foundation [Grant 2022B1515020082].

Mond-Weir and Wolfe Duality of Set-Valued Fractional Minimax Problems in Terms of Contingent Epi-Derivative of Second-Order

This paper is devoted to provide sufficient Karush Kuhn Tucker (in short, KKT) conditions of optimality of second-order for a set-valued fractional minimax problem. In addition, we define duals of the types Mond-Weir and Wolfe of second-order for the problem. Further we obtain the theorems of duality under contingent epi-derivative together with generalized cone convexity suppositions of second-order.

Large-Scale Least Squares Twin SVMs

In the last decade, twin support vector machine (TWSVM) classifiers have achieved considerable emphasis on pattern classification tasks. However, the TWSVM formulation still suffers from the following two shortcomings: (1) TWSVM deals with the inverse matrix calculation in the Wolfe-dual problems, which is intractable for large-scale datasets with numerous features and samples, and (2) TWSVM minimizes the empirical risk instead of the structural risk in its formulation. With the advent of huge amounts of data today, these disadvantages render TWSVM an ineffective choice for pattern classification tasks. In this article, we propose an efficient large-scale least squares twin support vector machine (LS-LSTSVM) for pattern classification that rectifies all the aforementioned shortcomings. The proposed LS-LSTSVM introduces different Lagrangian functions to eliminate the need for calculating inverse matrices. The proposed LS-LSTSVM also does not employ kernel-generated surfaces for the non-linear case, and thus uses the kernel trick directly. This ensures that the proposed LS-LSTSVM model is superior to the original TWSVM and LSTSVM. Lastly, the structural risk is minimized in LS-LSTSVM. This exhibits the essence of statistical learning theory, and consequently, classification accuracy on datasets can be improved due to this change. The proposed LS-LSTSVM is solved using the sequential minimal optimization (SMO) technique, making it more suitable for large-scale problems. We further proved the convergence of the proposed LS-LSTSVM. Exhaustive experiments on several real-world benchmarks and NDC-based large-scale datasets demonstrate that the proposed LS-LSTSVM is feasible for large datasets and, in most cases, performed better than existing algorithms.

Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

We propose a generalized second-order asymptotic contingent epiderivative of a set-valued mapping, study its properties, as well as relations to some second-order contingent epiderivatives, and sufficient conditions for its existence. Then, using these epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints. Both second-order necessary conditions and sufficient conditions for optimality of the Karush–Kuhn–Tucker type are established under the second-order constraint qualification. An application to Mond–Weir and Wolfe duality schemes is also presented. Some remarks and examples are provided to illustrate our results.

Duality results for interval-valued pseudoconvex optimization problem with equilibrium constraints with applications

This paper is devoted to constructing Wolfe and Mond–Weir dual models for interval-valued pseudoconvex optimization problem with equilibrium constraints, as well as providing weak and strong duality theorems for the same using the notion of contingent epiderivatives with pseudoconvex functions in real Banach spaces. First, we introduce the Mangasarian–Fromovitz type regularity condition and the two Wolfe and Mond–Weir dual models to such problem. Second, under suitable assumptions on the pseudoconvexity of objective and constraint functions, weak and strong duality theorems for the interval-valued pseudoconvex optimization problem with equilibrium constraints and its Mond–Weir and Wolfe dual problems are derived. An application of the obtained results for the GA-stationary vector to such interval-valued pseudoconvex optimization problem on sufficient optimality is presented. We also give several examples that illustrate our results in the paper.

Điều kiện tối ưu và đối ngẫu cho bài toán tối ưu đa trị sử dụng đạo hàm đa trị Clarke theo hướng nón

This paper is to deal with Mond-Weir duality and Wolfe duality for constrained set-valued optimization problems in terms of cone-directed Clarke derivatives. Firstly, necessary and sufficient optimality conditions for constrained set-valued optimizations in terms of cone-directed Clarke derivatives for the cone-semilocally convex like maps are investigated. Then, the Mond-Weir duality and Wolfe duality for a constrained set-valued optimization and their weak duality, strong duality and converse duality are considered.

Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions

This paper deals with convex semi-infinite programming with multiple interval-valued objective functions. We first investigate necessary and sufficient Karush–Kuhn–Tucker optimality conditions for some types of optimal solutions. Then, we formulate types of Mond–Weir and Wolfe dual problems and explore duality relations under convexity assumptions. Some examples are provided to illustrate the advantages of our results in some cases.

Quadratic unconstrained binary optimization formulation for rectified-linear-unit-type functions.

We propose a quadratic unconstrained binary optimization (QUBO) formulation of rectified-linear-unit (ReLU) type functions. Different from the q-loss function proposed by Denchev etal. [in Proceedings of the 29th International Conference on Machine Learning, Edinburgh, edited by J. Langford and J. Pineau (Omnipress, Madison, USA, 2012)], a simple discussion based on the Legendre duality is not sufficient to obtain the QUBO formulation of ReLU-type functions. In addition to the Legendre duality, we employ the Wolfe duality, and the QUBO formulation of ReLU type is derived. The QUBO formulation is available in Ising-type annealing methods, including quantum annealing machines.

Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.

Robust topology optimization under multiple independent unknown-but-bounded loads

The optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable, or even lead to catastrophic failures. A two-level optimization formulation is often used in the Robust Topology Optimization (RTO) under uncertain loads. Various approaches have been reported to identify the critical loads associated with the worst structure responses. Because Convex Model approaches apply convex approximations to the original non-convex model at the lower level, the optimal designs obtained by these methods are greatly dependent on the quality of the approximation. In this paper, a new formulation based on the Wolfe duality for the RTO problems with multiple independent unknown-but-bounded loads is proposed. Following the two-level formulation, the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the Wolfe duality. Both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non-convexity associated with the lower level formulation. Numerical examples are also presented to demonstrate the proposed approach.

Saddle point criteria and Wolfe duality in nonsmooth (Φ, ρ)-invex vector optimization problems with inequality and equality constraints

In this paper, saddle point criteria and Wolfe duality theorems are established for a new class of nondifferentiable vector optimization problems with inequality and equality constraints. The results are proved under nondifferentiable (Φ, ρ)-invexity and related scalar and vector-valued Lagrangians defined for the considered nonsmooth multiobjective programming problem. It turns out that the results are established for such vector optimization problems in which not all functions constituting a vector optimization problem possess the fundamental property of invexity and the most of generalized invexity notions previously defined in the literature.

Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes

In the first part of this paper series, a unified duality scheme for a constrained extremum problem is proposed by virtue of the image space analysis. In the present paper, we pay our attention to study of some special duality schemes. Particularly, the Lagrange-type duality, Wolfe duality and Mond---Weir duality are discussed as special duality schemes in a unified interpretation. Moreover, three practical classes of regular weak separation functions are also considered.

Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis

This paper is concerned with a unified duality theory for a constrained extremum problem. Following along with the image space analysis, a unified duality scheme for a constrained extremum problem is proposed by virtue of the class of regular weak separation functions in the image space. Some equivalent characterizations of the zero duality property are obtained under an appropriate assumption. Moreover, some necessary and sufficient conditions for the zero duality property are also established in terms of the perturbation function. In the accompanying paper, the Lagrange-type duality, Wolfe duality and Mond---Weir duality will be discussed as special duality schemes in a unified interpretation. Simultaneously, three practical classes of regular weak separation functions will be also considered.

Optimality Condition and Wolfe Duality for Invex Interval-Valued Nonlinear Programming Problems

The concepts of preinvex and invex are extended to the interval-valued functions. Under the assumption of invexity, the Karush-Kuhn-Tucker optimality sufficient and necessary conditions for interval-valued nonlinear programming problems are derived. Based on the concepts of having no duality gap in weak and strong sense, the Wolfe duality theorems for the invex interval-valued nonlinear programming problems are proposed in this paper.

Optimality and duality for proper and isolated efficiencies in multiobjective optimization

We use some advanced tools of variational analysis and generalized differentiation such as the nonsmooth version of Fermat’s rule, the limiting/Mordukhovich subdifferential of maximum functions, and the sum rules for the Fréchet subdifferential and for the limiting one to establish necessary conditions for (local) properly efficient solutions and (local) isolated minimizers of a multiobjective optimization problem involving inequality and equality constraints. Sufficient conditions for the existence of such solutions are also provided under assumptions of (local) convex/affine functions or L-invex-infine functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose a type of Wolfe dual problems and examine weak/strong duality relations under L-invexity-infineness hypotheses.

How to solve a semi-infinite optimization problem

After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. A section on recent genericity results includes a discussion of the symmetry effect in generalized semi-infinite optimization.

A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms.

Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

We consider nondifferentiable minimax fractional programming problems involving B-(p, r)-invex functions with respect to η and b. Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr B-(p, r)-invexity assumption on objective and constraint functions. Parametric duality, Mond-Weir duality, and Wolfe duality problems may be formulated, and duality results are derived under B-(p, r)-invex functions.