Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

Abstract

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.

Appendix: SDP Representations of Dual Programs

Finally, for the sake of completeness, we show how our dual problem $(\mathcal{D})$ given in (11) can be represented by a semidefinite linear programming problem. To this aim, let us recall some basic facts on the relationship between sum-of-squares polynomials and semidefinite programming problems.

We denote by $\mathbb{S}^{n}$ the space of symmetric n×n matrices. For any $A,B\in\mathbb{S}^{n}$, we write A⪰0 if and only if A is positive semidefinite, and 〈A,B〉 stands for $\operatorname{trace}(AB)$. Let $\mathbb{S}^{n}_{+}:=\{ A \in\mathbb {S}^{n} : A\succeq0\}$ be the closed and convex cone of positive semidefinite n×n (symmetric) matrices. The space of all real polynomials on $\mathbb{R}^{n}$ with degree d is denoted by $\mathbb{R}_{d}[x_{1},\ldots ,x_{n}]$ and its canonical basis is given by

$$ y(x) \equiv\bigl(x^{\alpha}\bigr)_{\vert \alpha \vert \leq d} := \bigl(1,x_1,x_2, \ldots,x_n,x_1^2,x_1x_2, \ldots,x_2^2,\ldots,x_n^2, \ldots,x_1^{d},\ldots,x_n^d \bigr)^T, $$

which has dimension $e(d,n):=\binom{n+d}{d}$, and $\alpha\in (\{0\} \cup\mathbb{N} )^{n}$ is a multi-index such that $\vert \alpha \vert :=\sum_{i=1}^{n}{\alpha_{i}}$. Let $\mathcal{N}:=\{ \alpha\in (\{0\}\cup \mathbb{N} )^{n} : \vert \alpha \vert \leq d \}$. Thus, if f is a polynomial on $\mathbb{R}^{n}$ with degree at most d, one has

$$f(x) = \sum_{\alpha\in\mathcal{N}} f_{\alpha}x^{\alpha}. $$

Assume that d is an even number, and let k:=d/2. Then, according to [7, Proposition 2.1], f is a sum-of-squares polynomial if and only if there exists $Q\in\mathbb{S}_{+}^{e(k,n)}$ such that f(x)=y(x)^T Q  y(x). By writing $y(x) y(x)^{T} = \sum_{\alpha\in\mathcal {N}} B_{\alpha}x^{\alpha}$ for appropriate matrices $(B_{\alpha })\subset\mathbb{S}^{e(k,n)}$, one finds that f is a sum-of-squares polynomial if and only if there exists $Q \in\mathbb{S}_{+}^{e(k,n)}$ such that 〈Q,B _α 〉=f _α for all $\alpha \in\mathcal{N}$.

Using the above characterization, we see that our dual problem $(\mathcal{D})$ can be equivalently rewritten as the following semidefinite programming problem:

$$ \begin{aligned} &(\mathcal{SD})\quad \sup \mu\\ &\phantom{(\mathcal{SD})\quad} \text{s.t.}\quad \sum_{j=1}^{r}{\delta_j (p_j)_{\alpha}} + \sum _{i=1}^{m}{\lambda_i (g_i)_{\alpha}} - \mu q_{\alpha} = \langle Q,B_{\alpha} \rangle, \quad\forall\alpha\in\mathcal{N}, \\ &\phantom{(\mathcal{SD})\quad \text{s.t.}\quad} \sum_{j=1}^{r}{\delta_j} = 1, \\ &\phantom{(\mathcal{SD})\quad\text{s.t.}\quad} \delta\in\mathbb{R}^r_+, \quad\lambda\in\mathbb{R}^m_+,\quad \mu\in\mathbb {R},\quad Q \in\mathbb{S}^{e(k,n)}_{+}. \end{aligned} $$

Letting q _α =1 for α=(0,…,0) and q _α =0 otherwise, we get the SDP representation for problem (D) in (3).