Interior Point Conditions Research Articles

Extension of primal-dual interior point method based on a kernel function for linear fractional problem

Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.

Implementation in Java of Algorithms for Semidefinite Optimization

Abstract We discuss the possibility of solving the semidefinite optimization problem using interior-point algorithms. We present the primal and dual semidefinite programming problems, and then determine the interior-point condition and the optimality criteria. We analyze the central path system and the modification of this, using the method of algebraically equivalent transformation. We use the Nesterov-Todd scaling technique to obtain the proper search directions. We give a modified version of the Nesterov-Todd step interior-point algorithm based on the implementation point of view. We present some numerical results based on a code implemented in the Java programming language. We compare the results obtained for the identity map and the square root function within the framework of the algebraically equivalent transformation technique.

Eigenfunction orthogonality for one-dimensional acoustic systems with interior or end point conditions.

Some common exercises presented in introductory acoustics courses and texts illustrate solutions involving eigenvalues and eigenfunctions. Challenging extensions of these, even for one-dimensional (1D) systems, might involve a mass or spring loading the acoustic medium at an end point or at an interior point. These problems might be extended further by requiring that some given function be expanded in a series of the eigenfunctions, but such extended problems may lead to unexpected complications in regard to eigenfunction orthogonality. In this paper, Sturm-Liouville theory is used to develop a systematic method for predetermining eigenfunction orthogonality for 1D systems loaded at end points or interior points or having properties that change with jump discontinuities.

Duality Theorems for Convex and Quasiconvex Set Functions

In mathematical programming, duality theorems play a central role. Especially, in convex and quasiconvex programming, Lagrange duality and surrogate duality have been studied extensively. Additionally, constraint qualifications are essential ingredients of the powerful duality theory. The best-known constraint qualifications are the interior point conditions, also known as the Slater-type constraint qualifications. A typical example of mathematical programming is a minimization problem of a real-valued function on a vector space. This types of problems have been studied widely and have been generalized in several directions. Recently, the authors investigate set functions and Fenchel duality. However, duality theorems and its constraint qualifications for mathematical programming with set functions have not been studied yet. It is expected to study set functions and duality theorems. In this paper, we study duality theorems for convex and quasiconvex set functions. We show Lagrange duality theorem for convex set functions and surrogate duality theorem for quasiconvex set functions under the Slater condition. As an application, we investigate an uncertain problem with motion uncertainty.

Spectral distribution problem for Sturm-Liouville with operators the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter

Spectral distribution problem for Sturm-Liouville with operators the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter

Linear Regularity and Linear Convergence of Projection-Based Methods for Solving Convex Feasibility Problems

For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here.

Numerical Reconstruction of Potentials Based on a Sequence of Inverse Sturm–Liouville Problems

Numerical solutions to inverse Sturm–Liouville problems by three spectra, on a finite interval [Formula: see text] are considered. Dirichlet boundary conditions are assumed to hold at the end-points of the interval [Formula: see text], and either a Dirichlet or a Robin-type interior point condition is imposed at an arbitrary interior point [Formula: see text] of [Formula: see text]. We demonstrate that the numerical solution of the inverse three spectra problem corresponding to an interior point [Formula: see text] depends continuously on [Formula: see text]. This statement is particularly useful when measurements of the input data cannot be obtained at the interior point [Formula: see text], but they can be obtained at a nearby interior point. Consequently, we could solve approximately the original inverse problem by departing slightly from [Formula: see text]. We validate this statement through numerical experiments for sequences of interior points [Formula: see text] converging to [Formula: see text].

MIXING TIME OF A + B− > 0 IN ONE DIMENSION

A mixing time density of [Formula: see text] on a finite one-dimensional domain is defined for general initial and boundary conditions in which [Formula: see text] and [Formula: see text] diffuse at the same rate. The density is a measure of the number of [Formula: see text] and [Formula: see text] particles that mix through the center of the reaction zone. It also corresponds to the reaction density for the special case in which [Formula: see text] and [Formula: see text] annihilate upon contact. An exact expression is found for the generating function of the mixing time. The analysis is extended to multiple reaction fronts and finitely ramified fractals. The method involves using the kernel of the Laplace transform integral operator to map and analyze a moving homogeneous Dirichlet interior point condition.

On Constraint Qualification for an Infinite System of Quasiconvex Inequalities in Normed Linear Space

The constraint qualification Q-CCCQ plays an important role in quasiconvex programming and has been developed by many authors to investigate the set containment problem, duality and optimality conditions for quasiconvex programming. In this paper, we consider an infinite quasiconvex inequality system defined by a family of proper lower semicontinuous quasiconvex functions $\{h_i : i \in I \}$ and establish some sufficient conditions for ensuring the Q-CCCQ in terms of the interior-point condition together with approximate continuity assumption of the function $i \mapsto h_i(x)$.

Circumventing the Slater conundrum in countably infinite linear programs

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researchers for a few decades; it has recently been called the Slater conundrum. We find a way around this hurdle.We propose a pair of primal-dual spaces with three properties: the series in the primal and dual objective functions converge; the series defined by the rows and columns of the primal constraint matrix converge; and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. Our results are illustrated using several examples, and are applied to countable-state Markov decision processes and to a problem in robust optimization.

Duality in Countably Infinite Linear Programs

Duality results on countably infinite linear programs (CILPs) are scarce. Roughly speaking, one fundamental hurdle is as follows. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual problem where the constraints cannot be expressed using the ordinary transpose of the primal doubly-infinite constraint matrix. Subspaces that do allow us to write the dual using this ordinary transpose do not admit an interior point. This has recently been called the "Slater conundrum." We find a way around this hurdle. We propose a pair of primal-dual spaces with three properties: (i) the series in the primal and dual objective functions converge; (ii) the series defined by the rows and columns of the primal constraint matrix converge; and (iii) the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. We also prove that the Lagrangean of our CILPs provides a linear support to the primal optimal value function. Finally, we establish that saddle points of the Lagrangean are equivalent to complementary feasible solutions of our CILPs. Our results are illustrated using several examples, and are also applied to countable-state Markov decision processes and to a problem in robust optimization.

Regularity conditions via generalized interiority notions in convex optimization: New achievements and their relation to some classical statements

For the existence of strong duality in convex optimization regularity conditions play an indisputable role. In this article we mainly deal with regularity conditions formulated by means of different generalizations of the notion of interior of a set. The primal–dual pair we investigate is a general one expressed in the language of a perturbation function and by employing its Fenchel–Moreau conjugate. After providing an overview on the generalized interior-point conditions that exist in the literature we introduce several new ones formulated by means of the quasi-interior and quasi-relative interior. We underline the advantages of the new conditions vis-á-vis the classical ones and illustrate our investigations by numerous examples. We conclude this article by particularizing the general approach to the classical Fenchel and Lagrange duality concepts.

Exploring complexity of large update interior-point methods for [formula omitted] linear complementarity problem based on Kernel function

Interior-point methods not only are the most effective methods in practice but also have polynomial-time complexity. The large update interior-point methods perform in practice much better than the small update methods which have the best known theoretical complexity. In this paper, motivated by the complexity results for linear optimization based on kernel functions, we extend a generic primal-dual interior-point algorithm based on a new kernel function to solve P ∗ ( κ ) linear complementarity problems. By using some elegant and simple tools and having interior-point condition, we show that the large update primal-dual interior-point methods for solving P ∗ ( κ ) linear complementarity problems enjoys O q ( 1 + 2 κ ) n ( log n ) q + 1 q log n ε iteration bound which becomes O ( 1 + 2 κ ) n log n log ( log n ) log n ε with special choices of the parameter q. This bound is much better than the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization field. Some computational results have been provided.

BCQ CONDITION AND CHARACTERIZATION OF THE BEST UNIFORM APPROXIMATION WITH GENERALIZED RESTRICTIONS

The problem of the best uniform approximation with generalized restrictions is investigated.Under the assumption that l and u have finite intersection points,by intro- ducing the concepts of the sub-strong interior point condition and making use of the BCQ condition from optimization theory,we study the relationship between the sub-strong interior point condition,BCQ condition and the characterization of the best uniform approximation with generalized restrictions.

Regularity Conditions via Quasi-Relative Interior in Convex Programming

We give some new regularity conditions for Fenchel duality in separated locally convex vector spaces, written in terms of the notion of quasi interior and quasi-relative interior, respectively. We provide also an example of a convex optimization problem for which the classical generalized interior-point conditions given so far in the literature cannot be applied, while the one given by us is applicable. By using a technique developed by Magnanti, we derive some duality results for the optimization problem with cone constraints and its Lagrange dual problem, and we show that a duality result recently given in the literature for this pair of problems has self-contradictory assumptions.

Multiple constrained rivaling actuators in the optimal control of miniaturized manipulators

Miniaturized robotic manipulators are a key elements in future high-precision telesurgery with artificial fingers and hands. A major challenge is the development of suitable drives for the micro-joints, because those small drives are either slow or can produce only low specific forces. A possible solution are joints driven by weak, but fast, and strong, but slow, actuators acting in parallel. This results in an optimal control problem of a multibody system subject to rivaling controls. Not only the control amplitude, but also the rate of change of the control is constrained for the respective actuator. In optimal control theory, derivatives of controls are forbidden by definition. The straightforward approach—defining the derivatives of the controls as the new control variables—results in a very sensitive differential-algebraic control problem with a complicated structure. A new approach is proposed: The original control problem is converted into a piecewise defined nonlinear multi-point boundary value problem for ordinary differential equations with nonconstant dimension. Based on the minimum principle, a strict and careful mathematical analysis of the coupling of the single parts of the boundary value problem leads to new interior point conditions at the junction points. As an example, three-dimensional energy optimal trajectories are calculated for a six-sectional branched manipulator. Stable, efficient and very accurate numerical solutions are obtained by the new procedure.

A new condition for maximal monotonicity via representative functions

In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S + A ∗ T A , where S : X ⇉ X ∗ , T : Y ⇉ Y ∗ are two maximal monotone operators, A : X → Y is a linear continuous mapping and X , Y are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.

On best restricted range approximation in continuous complex-valued function spaces

To provide a Kolmogorov-type condition for characterizing a best approximation in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when some Ω t is a line-segment or a singleton. The main aim of the present paper is to remove this restriction by virtue of a study of the notion of the strong CHIP for an infinite system of closed convex sets in a continuous complex-valued function space.

Necessary and sufficient conditions for stable conjugate duality

The conjugate duality, which states that inf x ∈ X φ ( x , 0 ) = max v ∈ Y ′ - φ * ( 0 , v ) , whenever a regularity condition on φ is satisfied, is a key result in convex analysis and optimization, where φ : X × Y → R ∪ { + ∞ } is a convex function, X and Y are Banach spaces, Y ′ is the continuous dual space of Y and φ * is the Fenchel–Moreau conjugate of φ . In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf x ∈ X { φ ( x , 0 ) + x * ( x ) } = max v ∈ Y ′ { - φ * ( - x * , v ) } , ∀ x * ∈ X ′ and then obtain a new epigraph regularity condition for the conjugate duality. The regularity condition is shown to be much more general than the popularly known interior-point type conditions. As an easy consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel–Rockafellar duality theorem. In the case where one of the functions involved is a polyhedral convex function, we provide generalized interior-point conditions for the epigraph regularity condition. Moreover, we show that a stable Fenchel's duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds.

A new geometric condition for Fenchel's duality in infinite dimensional spaces

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the e-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality.