Reverse Convex Research Articles

A note on duality in reverse convex optimization

In this note we explore duality in reverse convex optimization with reverse convex inequality constraints. While we are examining the special case of a finite index set of inequality constraints, we are primarily interested in the general case of an arbitrary infinite index set with neither extra topological structure nor extra measure structure.

Around a Farkas type lemma

The first two authors of this paper asserted in Lemma 4 of ‘New Farkas-type constraint qualifications in convex infinite programming’ (DOI: 10.1051/cocv:2007027) that a given reverse convex inequality is consequence of a given convex system satisfying the Farkas–Minkowski constraint qualification if and only if certain set depending on the data contains a particular point of the vertical axis. This paper identifies a hidden assumption in this reverse Farkas lemma which always holds in its applications to nontrivial optimization problems. Moreover, it shows that the statement remains valid when the Farkas–Minkowski constraint qualification fails by replacing the mentioned set by its closure. This hidden assumption is also characterized in terms of the data. Finally, the paper provides some applications to convex infinite systems and to convex infinite optimization problems.

Duality and optimality conditions for reverse convex programs via a convex decomposition

Duality and optimality conditions for reverse convex programs via a convex decomposition

Necessary Optimality Conditions for Vector Reverse Convex Minimization Problems via a Conjugate Duality

Necessary Optimality Conditions for Vector Reverse Convex Minimization Problems via a Conjugate Duality

Intersection Disjunctions for Reverse Convex Sets

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multiterm disjunction by considering the convex set’s recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set.

On tackling reverse convex constraints for non-overlapping of unequal circles

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

Optimal Relay Placement in Magnetic Induction-Based Internet of Underwater Things

The advances in developing the Internet of Underwater Things (IoUT) can lead to numerous applications, namely, environmental monitoring, underwater navigation, and surveillance. Magnetic Induction (MI) is a promising communication solution for IoUT networks. In this article, we focus on a deep-ocean monitoring network, where the underwater relays communicate using MI induction. Placement of the relays in such a system is an exciting and challenging task. Therefore, we propose an optimal relay placement solution for relays in MI-based IoUT networks to improve network throughput. First, we formulate the relay placement problem as a non-convex optimization problem. Then, we propose a global optimization technique based on Reverse Convex Programming (RCP) to solve the non-convex function. Finally, we compare the results of the proposed scheme with the uniform deployment to show its effectiveness.

The complexity results of the sparse optimization problems and reverse convex optimization problems

In this paper, we study complexity results of sparse optimization problems and reverse convex optimization problems. These problems are very important subjects of optimization problems. We prove that the complexity result of the sparsity constraint problem and sparse solution problem are all NP-hard in the strong sense and even testing feasibility of the sparsity constraint is NP-complete in the strong sense. Then the sparse optimization problem is NP-hard in the strong sense. We also prove that the reverse convex problem is NP-hard in the strong sense by transforming the sparsity constraint into a reverse convex constraint.

Solving nonconvex nonlinear programs with reverse convex constraints by sequential linear programming

AbstractThe sequential linear programming (SLP) method for solving nonlinear problems was introduced in the 1960s. Many papers that attempted to use SLP reported poor performance and convergence issues. We found that nonlinear programs with reverse convex constraints, which are the most difficult nonlinear programs with many local optima, are solved (heuristically) very well by SLP. We proved that for this type of problems, the solutions to the sequence of the linear programming problems converge to a local optimum. Since the final solution depends on the starting solution, we propose to apply SLP in a multistart approach starting from randomly generated solutions. This multistart SLP is very easy to implement. We recommend that the research community reconsiders the application of SLP for this type of problems.

Mixed-integer bilevel representability

We study the representability of sets by extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer variables exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets (up to topological closures) forms an algebra of sets; i.e., this family is closed under finite unions, intersections and complementation.

Solving Reverse Convex Programming Problems Using a Generalized Cutting-Plane Method

It is well known that each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is supremally generated by affine functions. More precisely, each convex function $$f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$$ is the upper envelope of its affine minorants. In this paper, we propose an algorithm for solving reverse convex programming problems by using such a representation together with a generalized cutting-plane method. Indeed, by applying this representation, we solve a sequence of problems with a smaller feasible set, in which the reverse convex constraint is replaced by a still reverse convex but polyhedral constraint. Moreover, we prove that the proposed algorithm converges, under suitable assumptions, to an optimal solution of the original problem. This algorithm is coded in MATLAB language and is evaluated by some numerical examples.

A customized branch-and-bound approach for irregular shape nesting

We study the Nesting Problem, which aims to determine a configuration of a set of irregular shapes within a rectangular sheet of material of fixed width, such that no overlap among the shapes exists, and such that the length of the sheet is minimized. When both translation and rotation of the shapes are allowed, the problem can be formulated as a nonconvex quadratically constrained programming model that approximates each shape by a set of inscribed circles and enforces that circle pairs stemming from different shapes do not overlap. However, despite many recent advances in today’s global optimization solvers, solving this nonconvex model to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we propose a customized branch-and-bound approach to address the Nesting Problem to guaranteed optimality. Our approach utilizes a novel branching scheme to deal with the reverse convex quadratic constraints in the quadratic model and incorporates a number of problem-specific algorithmic tweaks. Our computational studies on a suite of 64 benchmark instances demonstrate the customized algorithm’s effectiveness and competitiveness over the use of general-purpose global optimization solvers, including for the first time the ability to find global optimal nestings featuring five polygons under free rotation.

Asymptotic Farkas lemmas for convex systems

In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization

Robust Power-Splitting SWIPT Beamforming for Broadcast Channels

This letter considers the multiple-input single-output (MISO) broadcast system for simultaneous wireless information and power transfer (SWIPT) using receiver power splitting and aims to optimize jointly the beamforming vectors and the power splitting ratios for minimizing the transmit power of the base station (BS) subject to the individual signal-to-interference-plus-noise ratio (SINR) and the energy-harvesting constraints at the mobile stations (MSs). However, the CSI is assumed imperfect but has a deterministic uncertainty region. Unlike existing attempts that resort to iterations guided by semidefinite relaxation (SDR), we propose a reverse convex nonsmooth optimization algorithm, which provides the near-optimal rank-one solution.

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.

A Global Optimization Approach for Solving Generalized Nonlinear Multiplicative Programming Problem

This paper presents a global optimization algorithm for solving globally the generalized nonlinear multiplicative programming (MP) with a nonconvex constraint set. The algorithm uses a branch and bound scheme based on an equivalently reverse convex programming problem. As a result, in the computation procedure the main work is solving a series of linear programs that do not grow in size from iterations to iterations. Further several key strategies are proposed to enhance solution production, and some of them can be used to solve a general reverse convex programming problem. Numerical results show that the computational efficiency is improved obviously by using these strategies.

Feasible partition problem in reverse convex and convex mixed-integer programming

In this paper we consider the consistent partition problem in reverse convex and convex mixed-integer programming. In particular we will show that for the considered classes of convex functions, both integer and relaxed systems can be partitioned into two disjoint subsystems, each of which is consistent and defines an unbounded region. The polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided.

Numerically Simulating the Aerodynamic Interference Effect of Mean Aerodynamic Force Coefficients on Configurations of Bluff Bodies in Tandem

AbstractIn order to study the aerodynamic interference effects of single and multiple blunt bodies on the rectangular, reverse right angle, reverse inner convex arc, and reverse outer convex arc sections of serial bluff bodies, the finite volume method and the SIMPLE algorithm, the uniform viscous incompressible flow around blunt bodies at a subcritical Reynolds number was simulated by computational fluid dynamics (CFD) technology. The aerodynamic coefficient was analyzed, and the aerodynamic coefficients under various conditions (e.g., different wind angles of attack, chamfer dimensions, and cylinder spacing) were calculated. Results show that the drag and lift, as well as the torque coefficients, of each section of a single blunt decrease in accordance with the sequence of wind flow around the rectangular, reverse right angle, reverse inner convex arc, and reverse outer convex arc sections at different wind angles of attack. The drag and lift, as well as the torque coefficient, of the reverse right angl...

The Wiring Economy Principle for Designing Inference Networks

The wiring economy principle in neuroscience has explained many experimentally observed properties of neuronal networks by asserting the need to keep the axons and dendrites that connect neurons small in length. Just like neuronal networks, many distributed systems are physical constructs that incur deployment and maintenance costs for their communication infrastructure. Taking wiring economy as a design goal for engineering systems that perform distributed coordination and inference, this paper formulates and studies the tradeoff between performance and wiring cost. It is shown that separated communication topology design and physical node placement yields optimal design. Designing optimal networks is shown to be NP-complete. The natural relaxation to the integer network design problem is shown to be a reverse convex program. Small optimal networks are computed. Optimally placed random network topologies are demonstrated to have good performance.

Optimality conditions for DC programming problems with reverse convex constraints

In this paper, we give necessary and sufficient optimality conditions for DC programming problems with reverse convex constraints defined by a finite family of pseudoconvex functions. As application, we obtain necessary and sufficient optimality conditions for DC programming problems and fractional programming problems.