Slater Condition Research Articles

Linear Convergence of Subgradient Algorithm for Convex Feasibility on Riemannian Manifolds

We study the convergence issue of the subgradient algorithm for solving the convex feasibility problems in Riemannian manifolds, which was first proposed and analyzed by Bento and Melo [J. Optim. Theory Appl., 152 (2012), pp. 773--785]. The linear convergence property about the subgradient algorithm for solving the convex feasibility problems with the Slater condition in Riemannian manifolds are established, and some step sizes rules are suggested for finite convergence purposes, which are motivated by the work due to De Pierro Iusem [Appl. Math. Optim., 17 (1988), pp. 225--235]. As a by-product, the convergence result of this algorithm is obtained for the convex feasibility problem without the Slater condition assumption. These results extend and/or improve the corresponding known ones in both the Euclidean space and Riemannian manifolds.

Structure of Solutions for Continuous Linear Programs with Constant Coefficients

We consider continuous linear programs over a continuous finite time horizon $T$, with linear cost coefficient functions, linear right-hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In a recent paper, we have shown that under a Slater-type condition, these problems possess optimal strongly dual solutions. In this paper, we give a detailed description of optimal solutions and define a combinatorial analogue to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.

Quadratic minimization with portfolio and terminal wealth constraints

We address a problem of stochastic optimal control drawn from the area of mathematical finance. The goal is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio over the trading interval, together with a specified almost-sure lower-bound on the wealth at close of trade. We use a variational approach of Rockafellar which leads naturally to an appropriate vector space of dual variables, a dual functional on the space of dual variables such that the dual problem of maximizing the dual functional is guaranteed to have a solution (i.e. a Lagrange multiplier) when a simple and natural Slater condition holds for the terminal wealth constraint, and obtain necessary and sufficient conditions for optimality of a candidate wealth process. The dual variables are pairs, each comprising an Ito process paired with a member of the adjoint of the space of essentially bounded random variables measurable with respect to the event $$\sigma $$ -algebra at close of trade. The necessary and sufficient conditions are used to construct an optimal portfolio in terms of the Lagrange multiplier. The dual problem simplifies to maximization of a concave function over the real line when the portfolio is unconstrained but the terminal wealth constraint is maintained.

On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from Hogan (SIAM Rev 15:591–603, 1973a) for abstract feasible set mappings under complete convexity as well as standard differentiability results from Hogan (Oper Res 21:188–209, 1973b) for feasible set mappings in functional form under the Slater condition in the unfolded feasible set. In particular, we present sufficient conditions for the inner semi-continuity of feasible set mappings and, using techniques from nonsmooth analysis, provide functional descriptions of tangent cones to the domain of the optimal value function. The latter makes the stated directional differentiability results accessible for practical applications.

Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

A Lagrangian Dual Approach to the Single-Source Localization Problem

The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [“Exact and Approximate Solutions of Source Localization Problems,” IEEE Trans. Signal Process., vol. 56, pp. 1770-1778, 2008]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver for the multiple source localization problem on the tested cases.

An Approximate Dual Subgradient Algorithm for Multi-Agent Non-Convex Optimization

We consider a multi-agent optimization problem where agents subject to local, intermittent interactions aim to minimize a sum of local objective functions subject to a global inequality constraint and a global state constraint set. In contrast to previous work, we do not require that the objective, constraint functions, and state constraint sets are convex. In order to deal with time-varying network topologies satisfying a standard connectivity assumption, we resort to consensus algorithm techniques and the Lagrangian duality method. We slightly relax the requirement of exact consensus, and propose a distributed approximate dual subgradient algorithm to enable agents to asymptotically converge to a pair of primal-dual solutions to an approximate problem. To guarantee convergence, we assume that the Slater's condition is satisfied and the optimal solution set of the dual limit is singleton. We implement our algorithm over a source localization problem and compare the performance with existing algorithms.

Convergence results for a self-dual regularization of convex problems

We study a one-parameter regularization technique for convex optimization problems whose main feature is self-duality with respect to the Legendre–Fenchel conjugation. The self-dual technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one. For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater's condition.

Strong duality in robust semi-definite linear programming under data uncertainty

This article develops the deterministic approach to duality for semi-definite linear programming problems in the face of data uncertainty. We establish strong duality between the robust counterpart of an uncertain semi-definite linear programming model problem and the optimistic counterpart of its uncertain dual. We prove that strong duality between the deterministic counterparts holds under a characteristic cone condition. We also show that the characteristic cone condition is also necessary for the validity of strong duality for every linear objective function of the original model problem. In addition, we derive that a robust Slater condition alone ensures strong duality for uncertain semi-definite linear programs under spectral norm uncertainty and show, in this case, that the optimistic counterpart is also computationally tractable.

On Distributed Convex Optimization Under Inequality and Equality Constraints

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.

Dual Convergence for Penalty Algorithms in Convex Programming

Algorithms for convex programming, based on penalty methods, can be designed to have good primal convergence properties even without uniqueness of optimal solutions. Taking primal convergence for granted, in this paper we investigate the asymptotic behavior of an appropriate dual sequence obtained directly from primal iterates. First, under mild hypotheses, which include the standard Slater condition but neither strict complementarity nor second-order conditions, we show that this dual sequence is bounded and also, each cluster point belongs to the set of Karush–Kuhn–Tucker multipliers. Then we identify a general condition on the behavior of the generated primal objective values that ensures the full convergence of the dual sequence to a specific multiplier. This dual limit depends only on the particular penalty scheme used by the algorithm. Finally, we apply this approach to prove the first general dual convergence result of this kind for penalty-proximal algorithms in a nonlinear setting.

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing

Proving verification theorems can be tricky for models with both optimal stopping and state constraints. We pose and solve two alternative models of optimal consumption and investment with an optimal retirement date (optimal stopping) and various wealth constraints (state constraints). The solutions are parametric in closed form up to at most a constant. We prove the verification theorem for the main case with a nonnegative wealth constraint by combining the dynamic programming and Slater condition approaches. One unique feature of the proof is the application of the comparison principle to the differential equation solved by the proposed value function. In addition, we also obtain analytical comparative statics.

Distributed Primal–Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms

This paper studies the problem of optimizing the sum of multiple agents' local convex objective functions, subject to global convex inequality constraints and a convex state constraint set over a network. Through characterizing the primal and dual optimal solutions as the saddle points of the Lagrangian function associated with the problem, we propose a distributed algorithm, named the distributed primal-dual subgradient method, to provide approximate saddle points of the Lagrangian function, based on the distributed average consensus algorithms. Under Slater's condition, we obtain bounds on the convergence properties of the proposed method for a constant step size. Simulation examples are provided to demonstrate the effectiveness of the proposed method.

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.

A complete characterization of strong duality in nonconvex optimization with a single constraint

We first establish sufficient conditions ensuring strong duality for cone constrained nonconvex optimization problems under a generalized Slater-type condition. Such conditions allow us to cover situations where recent results cannot be applied. Afterwards, we provide a new complete characterization of strong duality for a problem with a single constraint: showing, in particular, that strong duality still holds without the standard Slater condition. This yields Lagrange multipliers characterizations of global optimality in case of (not necessarily convex) quadratic homogeneous functions after applying a generalized joint-range convexity result. Furthermore, a result which reduces a constrained minimization problem into one with a single constraint under generalized convexity assumptions, is also presented.

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.

On Implicit Active Constraints in Linear Semi-Infinite Programs with Unbounded Coefficients

The concept of implicit active constraints at a given point provides useful local information about the solution set of linear semi-infinite systems and about the optimal set in linear semi-infinite programming provided the set of gradient vectors of the constraints is bounded, commonly under the additional assumption that there exists some strong Slater point. This paper shows that the mentioned global boundedness condition can be replaced by a weaker local condition (LUB) based on locally active constraints (active in a ball of small radius whose center is some nominal point), providing geometric information about the solution set and Karush-Kuhn-Tucker type conditions for the optimal solution to be strongly unique. The maintaining of the latter property under sufficiently small perturbations of all the data is also analyzed, giving a characterization of its stability with respect to these perturbations in terms of the strong Slater condition, the so-called Extended-Nürnberger condition, and the LUB condition.

Optimal Control for an Elliptic System with Polygonal State Constraints

This work is devoted to stationary optimal control problems with polygonal constraints on the components of the state. Existence of Lagrange multipliers, of different regularities, is verified for the cases with and without Slater condition holding. For the numerical realization a semismooth Newton method is proposed for an appropriately chosen family of regularized problems. The asymptotic behavior of the regularized problem class is studied, and numerical feasibility of the method is shown.

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x 0 of the convex SIP problem by such conditions for x 0 in some nonlinear programming (NLP) problem denoted by NLP(I(x 0)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x 0)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.

AN INTERVIEW WITH HIROFUMI UZAWA

AN INTERVIEW WITH HIROFUMI UZAWA