On boundedness of (quasi-)convex integer optimization problems

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained integer optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or quasi-convex polynomial function over the feasible set contained in , and defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities. The conditions for boundedness are provided in the form of an implementable algorithm, terminating after a finite number of iterations, showing that for the considered class of functions, the integer programming problem with nonempty feasible region is unbounded if and only if the associated continuous optimization problem is unbounded. We also prove that for a broad class of objective functions (which in particular includes polynomials with integer coefficients), an optimal solution set of the constrained integer problem is nonempty over any subset of .     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Conditions for boundedness in concave programming under reverse convex and convex constraints

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.     

Existence of Global Minima for Constrained Optimization

We present a unified approach to establishing the existence of global minima of a (non)convex constrained optimization problem. Our results unify and generalize previous existence results for convex and nonconvex programs, including the Frank-Wolfe theorem, and for (quasi) convex quadratically constrained quadratic programs and convex polynomial programs. For example, instead of requiring the objective/constraint functions to be constant along certain recession directions, we only require them to linearly recede along these directions. Instead of requiring the objective/constraint functions to be convex polynomials, we only require the objective function to be a (quasi)convex polynomial over a polyhedral set and the constraint functions to be convex polynomials or the composition of coercive functions with linear mappings.We thank Professor Dimitri Bertsekas for his comments and support in the writing of this paper.     

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.     

A new class of alternative theorems for SOS-convex inequalities and robust optimization

In this paper, we present a new class of alternative theorems for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the class of alternative theorems. Consequently, we show that the optimal values of various classes of robust convex optimization problems are equal to the optimal values of related semidefinite programming problems (SDPs) and so, the value of the robust problem can be found by solving a single SDP. The class of problems includes programs with SOS-convex polynomials under data uncertainty in the objective function such as uncertain quadratically constrained quadratic programs. The SOS-convexity is a computationally tractable relaxation of convexity for a real polynomial. We also provide an application of our theorem of the alternative to a multi-objective convex optimization under data uncertainty.     

A new method for interpolating in a convex subset of a Hilbert space

In this paper, interpolating curve or surface with linear inequality constraints is considered as a general convex optimization problem in a Reproducing Kernel Hilbert Space. The aim of the present paper is to propose an approximation method in a very general framework based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. Numerical examples are provided to illustrate this result in the case of boundedness and monotonicity constraints in one and two dimensions.     

Solution existence and stability of quadratically constrained convex quadratic programs

We propose verifiable necessary and sufficient conditions for the solution existence of a convex quadratic program whose constraint set is defined by finitely many convex linear-quadratic inequalities. A related stability analysis of the problem is also considered.     

Minimal infeasible constraint sets in convex integer programs

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2 ⁿ , this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2 ⁿ −1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.     

A Frank–Wolfe Type Theorem for Convex Polynomial Programs

In 1956, Frank and Wolfe extended the fundamental existence theorem of linear programming by proving that an arbitrary quadratic function f attains its minimum over a nonempty convex polyhedral set X provided f is bounded from below over X. We show that a similar statement holds if f is a convex polynomial and X is the solution set of a system of convex polynomial inequalities. In fact, this result was published by the first author already in a 1977 book, but seems to have been unnoticed until now. Further, we discuss the behavior of convex polynomial sets under linear transformations and derive some consequences of the Frank–Wolfe type theorem for perturbed problems.     

Quasi-convex functions and applications to optimality conditions in nonlinear programming

We establish necessary and sufficient optimality conditions for quasi-convex programming. First, we treat some properties of the normal cone to the level set of a lower semicontinuous quasi-convex function defined on a Banach space. Next, we get our conditions via some variational inequalities.     

Integer Optimization on Convex Semialgebraic Sets

Let Y be a convex set in \Re ^k defined by polynomial inequalities and equations of degree at most d ≥ 2 with integer coefficients of binary length at most l . We show that if the set of optimal solutions of the integer programming problem min is not empty, then the problem has an optimal solution of binary length ld ^O(k4) . For fixed k , our bound implies a polynomial-time algorithm for computing an optimal integral solution y ^* . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming. Received August 3, 1998, and in revised form March 22, 1999.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Feasibility in reverse convex mixed-integer programming

In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility (CF), that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

Perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems

In this paper several types of perturbations on a convex inequality system are considered, and conditions are obtained for the system to be well-conditioned under these types of perturbations, where the well-conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations. It is shown that certain types of perturbations can be used to characterize the well-conditionedness of a convex inequality system, in which either the system has a bounded solution set and satisfies the Slater condition or an associated convex inequality system, which defines the recession cone of the solution set for the system, satisfies the Slater condition. Finally, sufficient conditions are given for the existence of a global error bound for an analytic system. It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function.     

Monotone operator theory in convex optimization

In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.     

Duality and existence theory for nondifferentiable programming

Necessary and sufficient conditions of optimality are given for a nonlinear nondifferentiable program, where the constraints are defined via closed convex cones and their polars. These results are then used to obtain an existence theorem for the corresponding stationary point problem, under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the programming problem. Furthermore, a dual problem is defined, and a strong duality theorem is obtained under the assumption that the constraint sets of the primal and dual problems are nonempty.     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.