Kuhn-Tucker conditions for mathematical programming problems in Banach spaces partially ordered by cone with empty interior are obtained under strong simultaneity condition. If partial ordered cone has interior point, it is proved that Slater and strong simultaneity conditions are equivalent.
We provide a simple and short proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints. The proof relies on an elementary linear algebra lemma and the local inverse theorem.
In the recent paper of Giorgi, Jiménez and Novo (J Optim Theory Appl 171:70--89, 2016), the authors introduced the so-called approximate Karush-Kuhn-Tucker (AKKT) condition for smooth multiobjective optimization problems and obtained some AKKT-type necessary optimality conditions and sufficient optimality conditions for weak efficient solutions of such a problem. In this note, we extend these optimality conditions to locally Lipschitz multiobjective optimization problems using Mordukhovich subdifferentials. Furthermore, we prove that, under some suitable additional conditions, an AKKT conditio…
In the present paper, we focus on the vector optimization problems with inequality constraints, where objective functions and constrained functions are Fréchet differentiable, and whose gradient mapping is locally Lipschitz on an open set. By using the second-order symmetric subdifferential and the second-order tangent set, we propose two types of second-order regularity conditions in the sense of Abadie. Then we establish some strong second-order Karush--Kuhn--Tucker necessary optimality conditions for Geoffrion properly efficient solutions of the considered problem. Examples are given to ill…
The Karush-Kuhn-Tucker and value function (lower-level value function, to be precise) reformulations are the most common single-level transformations of the bilevel optimization problem. So far, these reformulations have either been studied independently or as a joint optimization problem in an attempt to take advantage of the best properties from each model. To the best of our knowledge, these reformulations have not yet been compared in the existing literature. This paper is a first attempt towards establishing whether one of these reformulations is best at solving a given class of the optim…
Second-Order Karush-Kuhn-Tucker Optimality Conditions for Vector Problems with Continuously Differentiable Data and Second-Order Constraint Qualifications
This expository paper contains a concise introduction to some significant works concerning the Karush-Kuhn-Tucker condition, a necessary condition for a solution in local optimality in problems with equality and inequality constraints. The study of this optimality condition has a long history and culminated in the appearance of subdifferentials. The 1970s and early 1980s were important periods for new developments and various generalizations of subdifferentials were introduced, including the Clarke subdifferential and Demyanov-Rubinov quasidifferential. In this paper, we mainly present four ge…
With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates…
Although the Karush-Kuhn-Tucker conditions suggest a connection between a conic optimization problem and a complementarity problem, it is difficult to find an accessible explicit form of this relationship in the literature. This note will present such a relationship.
This paper deals with approximate solutions of an optimization problem with interval-valued objective function. Four types of approximate solution concepts of the problem are proposed by considering the partial ordering $LU$ on the set of all closed and bounded intervals. We show that these solutions exist under very weak conditions. Under suitable constraint qualifications, we derive Karush--Kuhn--Tucker necessary and sufficient optimality conditions for convex interval-valued optimization problems.
In this article we consider a convex feasible set described by inequality constraints that are continuous and not necessarily Lipschitz or convex. We show that if the Slater constraint qualification and a non-degeneracy condition are satisfied, then the Karush-Kuhn-Tucker type optimality condition is both necessary and sufficient. In this way we extend previous results which are proved for Lipschitz and differentiable inequalities.
The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization problem with finitely many convex objective functions and infinitely many convex constraints. To do this, we introduce new and already known data qualifications (conditions involving the constraints and/or the objectives) in order to get optimality conditions which are expressed in terms of either Karusk-Kuhn-Tucker multipliers or a new gap function associated with the given problem.
In this paper, we introduce a kind of approximate Karush--Kuhn--Tucker condition (AKKT) for a smooth cone-constrained vector optimization problem. We show that, without any constraint qualification, the AKKT condition is a necessary for a local weak efficient solution of the considered problem. For convex problems, we prove that the AKKT condition is a necessary and sufficient optimality condition for a global weak efficient solution. We also introduce some strict constraint qualifications associated with the AKKT condition.
By applying the perturbation function approach, we propose the Lagrangian and the conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the $Φ$-convexity theory and minimax theorems for $Φ$-convex functions. We provide conditions ensuring zero duality gap and introduce $Φ$-Karush-Kuhn-Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature.
In this paper, we study second-order necessary and sufficient optimality conditions of Karush--Kuhn--Tucker-type for locally optimal solutions in the sense of Pareto to a class of multi-objective optimal control problems with mixed pointwise constraint. To deal with the problems, we first derive second-order optimality conditions for abstract multi-objective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct, self-contained without using scalarization techniques.
This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimizatio…
Given a non-convex optimization problem, we study conditions under which every Karush-Kuhn-Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n-dimensions and show that these conditions become sufficient in the two-dimensional case. As an application of our results, we study the Optimal Power Flow problem, showing that under mild assumptions on the variable's bounds, our new necessary a…
The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming (NLSDP) is established by assuming the second-order sufficient condition (SOSC) and the …
Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush--Kuhn--Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying…
In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of "no-gap" second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared…
This paper is devoted to studying the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed f…
In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at the referee point. Some second-order constraint qualifications of Zangwill type, Abadie type and Mangasarian~--~Fromovitz type as well as a regularity condition of Abadie type are proposed in a nonsmooth setting. The connections between these proposed conditions are established. They are applied to develop second-order Karush--Kuhn--Tucker necessary optimality conditions for local (weak, Geoffrion properly) effic…
We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush-Kuhn-Tucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under whi…
Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimizations. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessa…
On Convex optimization without convex representation
本文考虑凸优化问题P：min {f（x）：x in K}，其中f是凸连续可微函数，K是Rn中的紧致凸集，对连续可微函数（g_j）有表示{x：g_j（x）>=0，j=1，;;，m}.我们讨论的情况下，g_j的不都是凹的（与凸规划，他们都是）。特别地，即使g_j不是凹函数，我们考虑与P相关联的具有参数μ的对数障碍函数phi_μ，其通常定义为凹函数（g_j）。证明了φ_μ，μ->0的任意平稳点序列（x_μ）的任意极限点是问题P的Karush-Kuhn-Tucker点，也是f在K上的全局极小点.
Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper de…
In this paper, we consider a nonlinear semi-infinite program that minimizes a function including a log-determinant (logdet) function over positive definite matrix constraints and infinitely many convex inequality constraints, called SIPLOG for short. The main purpose of the paper is to develop an algorithm for computing a Karush-Kuhn-Tucker (KKT) point for the SIPLOG efficiently. More specifically, we propose an interior point sequential quadratic programming-type method that inexactly solves a sequence of semi-infinite quadratic programs approximating the SIPLOG. Furthermore, to generate a se…
The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, which are based on the Approximate-Karush-Kuhn-Tucker necessary optimality condition and on the application of the reduction approach. Our definitions are strictly weaker than Robins…
Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas. We focus on the stability of constrained minimax optimization problems based on the notion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classical Jacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $C^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which does not require the st…
An effective means for analyzing the impact of novel operating schemes on power systems is time domain simulation, for example for investigating optimization-based curtailment of renewables to alleviate voltage violations. Traditionally, interior-point methods are used for solving the non-convex AC optimal power flow (OPF) problems arising in this type of simulation. This paper presents an alternative algorithm that better suits the simulation framework, because it can more effectively be warm-started, has linear computational and memory complexity in the problem size per iteration and globall…
In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Unlike the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems inexactly, and each QSDP is feasible. One more remarkable point of the proposed method is that constraint qualifications (CQs) or boundedness of Lagrange …
It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the problem has a solution. Another major result of this paper says that DCA sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably-chosen neighborhood of it. To deal with the implicitly defined iterative sequences, a local error bound for affine variational inequalities and nove…
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…
Due to the possible lack of primal-dual-type error bounds, the superlinear convergence for the Karush-Kuhn-Tucker (KKT) residues of the sequence generated by augmented Lagrangian method (ALM) for solving convex composite conic programming (CCCP) has long been an outstanding open question. In this paper, we aim to resolve this issue by first conducting convergence rate analysis for the ALM with Rockafellar's stopping criteria under only a mild quadratic growth condition on the dual of CCCP. More importantly, by further assuming that the Robinson constraint qualification holds, we establish the …
In this paper, we provide a complete characterization on the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for convex constrained optimization problems regularized by the nuclear norm function. This study is motivated by the recent work in [8], where the authors show that under the Robinson constraint qualification at a local optimal solution, the KKT solution mapping for a wide class of conic programming problem is robustly isolated calm if and only if both the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) are sati…
