Strong Duality Property Research Articles

Data-driven economic MPC with asymptotic stability and strong duality verification using Hankel matrix

We consider the problem of dynamic regulation with an economic cost function to control unknown linear systems, in which improving the economic performance and guaranteeing the stability of economical optimal equilibrium point are control objectives. A data-driven economic MPC scheme is presented using measured input-output trajectories without a prior system identification step. Our method uses Hankel matrices which include one input-output data trajectory for prediction in economic MPC, while persistently exciting of the input generating the data is needed. One of the novelties of the presented framework is directly verifying the strong duality property from input-output trajectory with the general cost function, considered as the supply rate. This is used to find a Lyapunov function for data-driven economic MPC. Under the strong duality assumption, asymptotic stability of the economical optimal equilibrium point for the closed-loop system with terminal equality constraint is guaranteed. The proposed data-driven economic MPC approach needs only persistently exciting data trajectory along with an upper bound on the system order and need no model description and no online parameter estimation. The proposed scheme applicability compared to the existing model-based economic MPC and data-driven MPC is illustrated for continuous stirred tank reactor (CSTR) and a numerical example and the robustness of the proposed scheme is evaluated in the case of measurement noise, as well as nonlinear model for CSTR system.

On Mond–Weir-Type Robust Duality for a Class of Uncertain Fractional Optimization Problems

This article is focused on the investigation of Mond–Weir-type robust duality for a class of semi-infinite multi-objective fractional optimization with uncertainty in the constraint functions. We first establish a Mond–Weir-type robust dual problem for this fractional optimization problem. Then, by combining a new robust-type subdifferential constraint qualification condition and a generalized convex-inclusion assumption, we present robust ε-quasi-weak and strong duality properties between this uncertain fractional optimization and its uncertain Mond–Weir-type robust dual problem. Moreover, we also investigate robust ε-quasi converse-like duality properties between them.

Approximate solutions for robust multiobjective optimization programming in Asplund spaces

In this paper, we study a nonsmooth/nonconvex multiobjective optimization problem with uncertain constraints in arbitrary Asplund spaces. We first provide necessary optimality condition in a fuzzy form for approximate weakly robust efficient solutions and then establish necessary optimality theorem for approximate weakly robust quasi-efficient solutions of the problem in the sense of the limiting subdifferential by exploiting a fuzzy optimality condition in terms of the Fréchet subdifferential. Sufficient conditions for approximate (weakly) robust quasi-efficient solutions to such a problem are also driven under the new concept of generalized pseudo convex functions. Finally, we address an approximate Mond-Weir-type dual robust problem to the reference problem and explore weak, strong, and converse duality properties under assumptions of pseudo convexity.

A Study on the Strong Duality of Second-Order Conic Relaxation of AC Optimal Power Flow in Radial Networks

For the popular second-order conic program (SOCP) formulation of AC optimal power flow (OPF) in a radial network, this paper first shows that it does not have the strong duality property in general. Then, through a series of restrictive reformulations, we derive a set of closed-form sufficient conditions on network parameters that ensure its strong duality. Numerical studies on IEEE 33-bus, 69-bus test networks and two real-world distribution systems confirm that non-negligible duality gaps <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">do exist</i> in this SOCP formulation, and also demonstrate the validity of the proposed sufficient conditions on closing the duality gap. Our results provide an analytical tool to ensure the strong duality of the SOCP power flow formulation and to support algorithm developments for its complex extensions.

Saddle point optimality criteria and duality for convex continuous-time programming problem

In this paper, convex continuous-time programming problem with inequality type of constraints is considered. We derive new saddle point optimality conditions and classical duality results such as weak and strong duality properties, under additional regularity assumption. A fundamental tool, employed in the derivation of the necessary saddle point optimality criteria and strong duality result for convex continuous-time programming, is a new version of a theorem of the alternative in infinite-dimensional spaces.

Convergence of Lasserre’s hierarchy: the general case

Lasserre’s moment-SOS hierarchy consists in approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming problems. Due to the generality of the initial GMP, applications of this technology are countless, and one can cite among them the polynomial optimization problem, the optimal control problem, the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations. The solution to the original GMP is then approximated with finite truncatures of its moment sequence. For each application, proving convergence of these truncatures towards the optimal moment sequence gives valuable insight on the problem, including convergence of the relaxed values to the original GMP’s optimal value. This note proposes a general proof of such convergence, regardless the problem one is faced with, under simple standard assumptions. As a byproduct of this proof, one also obtains strong duality properties both in the infinite dimensional GMP and its finite dimensional relaxations.

A primal–dual penalty method via rounded weighted-ℓ1 Lagrangian duality

We propose a new duality scheme based on a sequence of smooth minorants of the weighted- penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve non-convex constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) the sequence of dual problems is convex and (ii) the dual values monotonically increase to the optimal primal value. We use these properties to devise a subgradient based primal–dual method, and show that the generated primal sequence accumulates at a solution of the original problem. We illustrate the performance of the new method with three different types of test problems: A polynomial non-convex problem, large-scale instances of the celebrated kissing number problem, and the Markov–Dubins problem. Our numerical experiments demonstrate that, when compared with the traditional implementation of a well-known smooth solver, our new method (using the same well-known solver in its subproblem) can find better quality solutions, i.e. ‘deeper’ local minima, or solutions closer to the global minimum. Moreover, our method seems to be more time efficient, especially when the problem has a large number of constraints.

Linear semidefinite programming problems: regularisation and strong dual formulations

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

Weighted thresholding homotopy method for sparsity constrained optimization

We propose in this paper a novel weighted thresholding method for the sparsity-constrained optimization problem. By reformulating the problem equivalently as a mixed-integer programming, we investigate the Lagrange duality with respect to an $$l_1$$-norm constraint and show the strong duality property. Then we derive a weighted thresholding method for the inner Lagrangian problem, and analyze its convergence. In addition, we give an error bound of the solution under some assumptions. Further, based on the proposed method, we develop a homotopy algorithm with varying sparsity level and Lagrange multiplier, and prove that the algorithm converges to an L-stationary point of the primal problem under some conditions. Computational experiments show that the proposed algorithm is competitive with state-of-the-art methods for the sparsity-constrained optimization problem.

Dual Models and Return Allocations for Consensus Building Under Weighted Average Operators

Recent studies have examined minimum cost consensus models from the moderator’s perspective. However, current dual models aimed at deriving a maximum return have been limited to special cases. A further problem has been the return allocation to the individuals. With this in mind, this article presents two general dual consensus models, the first of which is a minimum cost consensus model, and the second of which allows the different individuals to have different tolerance levels. In both models the individual opinions and the group opinion are connected using a weighted average operator. To understand the economic significance of these models, some properties, such as the weak duality property, the optimality property, and the strong duality property are given. The relationships between the individual opinions, the unit costs, and the unit returns are also proven, and the core return allocation problem is addressed. It was found that in both cases, the costs the moderator needed to pay to a given individual were precisely equal to the returns the individual deserved. Numerical case studies were used to illustrate the proposed models, with the simulation results providing insights into the practical use of these optimization consensus models.

Distributed Optimization Framework for In-Network Data Processing

In-Network Processing (INP) is an effective way to aggregate and process data from different sources and forward the aggregated data to other nodes for further processing until it reaches the end user. There is a trade-off between energy consumption for processing data and communication energy spent on transferring the data. An essential requirement in the INP process is to ensure that the user expectation of quality of information (QoI) is delivered during the process. Using wireless sensor networks for illustration and with the aim of minimizing the total energy consumption of the system, we study and formulate the trade-off problem as a nonlinear optimization problem where the goal is to determine the optimal data reduction rate, while satisfying the QoI required by the user. The formulated problem is a Signomial Programming (SP) problem, which is a non-convex optimization problem. We propose two solution frameworks. First, we introduce an equivalent problem which is still SP and non-convex as the original one, but we prove that the strong duality property holds, and propose an efficient distributed algorithm to obtain the optimal data reduction rates, while delivering the required QoI. The second framework applies to the system with identical nodes and parameter settings. In such cases, we prove that the complexity of the problem can be reduced logarithmically. We evaluate our proposed frameworks under different parameter settings and illustrate the validity and performance of the proposed techniques through extensive simulation.

Lagrange Dual Method for Sparsity Constrained Optimization

In this paper, we investigate the $l_{0}$ quasi-norm constrained optimization problem in the Lagrange dual framework and show that the strong duality property holds. Motivated by the property, we propose a Lagrange dual method for the sparsity constrained optimization problem. The method adopts the bisection search technique to maximize the Lagrange dual function. For each Lagrange multiplier, we adopt the iterative hard thresholding method to minimize the Lagrange function. We show that the proposed method converges to an $L$ -stationary point of the primal problem. Computational experiments and comparisons on a number of test instances (including random compressed sensing instances and random and real sparse logistic regression instances) demonstrate the effectiveness of the proposed method in generating sparse solution accurately.

A note on linearized reformulations for a class of bilevel linear integer problems

We consider reformulations of a class of bilevel linear integer programs as equivalent linear mixed-integer programs (linear MIPs). The most common technique to reformulate such programs as a single-level problem is to replace the lower-level linear optimization problem by Karush–Kuhn–Tucker (KKT) optimality conditions. Employing the strong duality (SD) property of linear programs is an alternative method to perform such transformations. In this note, we describe two SD-based reformulations where the key idea is to exploit the binary expansion of upper-level integer variables. We compare the performance of an off-the-shelf MIP solver with the SD-based reformulations against the KKT-based one and show that the SD-based approaches can lead to orders of magnitude reduction in computational times for certain classes of instances.

Strong Duality and Dual Pricing Properties in Semi-Infinite Linear Programming: A non-Fourier–Motzkin Elimination Approach

Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang, recently published in Optimization, in this paper we show that the Fourier–Motzkin elimination is not needed in the study of the strong duality and dual pricing properties for semi-infinite programming. We also prove several new results on the strong duality and dual pricing properties. Specifically, we propose a new subspace, under which the strong duality property holds. We give a necessary and sufficient condition for the dual pricing property to hold under this subspace, which is further used to examine the examples presented in the Basu–Martin–Ryan paper.

A Duality Approach for Solving Control-Constrained Linear-Quadratic Optimal Control Problems

We use a Fenchel duality scheme for solving control-constrained linear-quadratic optimal control problems. We derive the dual of the optimal control problem explicitly, where the control constraints are embedded in the dual objective functional, which turns out to be continuously differentiable. We specifically prove that strong duality and saddle point properties hold. We carry out numerical experiments with the discretized primal and dual formulations of the problem, for which we implement powerful existing finite-dimensional optimization techniques and associated software. We illustrate that by solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved. Other numerical advantages are also discussed.

On semi-infinite minmax programming with generalized invexity

Here, we consider the minmax programming problem with a set of restrictions indexed in a compact. As a novelty, we obtain optimality criteria of the Kuhn--Tucker type involving a limited number of restrictions and prove both necessity and sufficiency under new weaker invexity assumptions. Also some dual problems are introduced and it is proved that the weak and strong duality properties hold within the same environment.

Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients

In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the corresponding duality problem in how far the solutions of the corresponding homogeneous systems determine the equations. For ordinary delay-differential (DD) equations these behavioral problems were investigated in a seminal paper by Glüsing-Lürssen (SIAM J Control Optim 35:480–499, 1997) and in later papers by Habets, Glüsing-Lürssen, Vettori and Zampieri. In these papers the delay-differential operators are considered as distributions with compact support which act on smooth functions or on arbitrary distributions via convolution. The entire analytic Laplace transforms of the distributions with compact support play an important part in the quoted papers. In our approach the partial differential-difference operators belong to various topological operator rings $$A$$ of holomorphic functions on subsets of $$\mathbb C ^n$$ and are thus studied in the frequency domain, the arguments of these (operator) functions being interpreted as generalized frequencies. We show that the topological duals $$A^{\prime }$$ of these operator rings with the canonical action of $$A$$ on $$A^{\prime }$$ have strong elimination and duality properties for $$_A A^{\prime }$$ -behaviors and admit concrete representations as spaces of analytic functions of systems theoretic interest. In particular, we study systems with generalized frequencies in the vicinity of suitable compact sets. An application to elimination for systems of periodic signals is given. We also solve an open problem of the quoted authors for DD-equations with incommensurate delays and analytic signals. Module theoretic methods in context with DD-equations have also been used by other authors, for instance by Fliess, Mounier, Rocha and Willems.

Duality for max-separable problems

In this paper we propose a general duality theory for a class of so called ‘max-separable’ optimization problems. In such problems functions h:Rk → R of the form h(x1, . . . , xk) = maxj hj(xj), occur both as objective functions and as constraint functions (hj are assumed to be strictly increasing functions of one variable). As a result we obtain pairs of max-separable optimization problems, which possess both weak and strong duality property without a duality gap.

A Lyapunov Function for Economic Optimizing Model Predictive Control

Standard model predictive control (MPC) yields an asymptotically stable steady-state solution using the following procedure. Given a dynamic model, a steady state of interest is selected, a stage cost is defined that measures deviation from this selected steady state, the controller cost function is a summation of this stage cost over a time horizon, and the optimal cost is shown to be a Lyapunov function for the closed-loop system. In this technical note, the stage cost is an arbitrary economic objective, which may not depend on a steady state, and the optimal cost is not a Lyapunov function for the closed-loop system. For a class of nonlinear systems and economic stage costs, this technical note constructs a suitable Lyapunov function, and the optimal steady-state solution of the economic stage cost is an asymptotically stable solution of the closed-loop system under economic MPC. Both finite and infinite horizons are treated. The class of nonlinear systems is defined by satisfaction of a strong duality property of the steady-state problem. This class includes linear systems with convex stage costs, generalizing previous stability results and providing a Lyapunov function for economic MPC or MPC with an unreachable setpoint and a linear model. A nonlinear chemical reactor example is provided illustrating these points.

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions

We apply the quadratic penalization technique to derive strong Lagrangian duality property for an inequality constrained invex program. Our results extend and improve the corresponding results in the literature.