FUNCHAL, PORTUGAL — Alberto João Jardim has won every election in Madeira — an archipelago famous for its robust wine — since 1978, shortly after Portugal’s return to democracy. Next Sunday, he is confident that voters will give him another four-year term as head of the regional government.

But while he has managed over the past three decades to turn the islands from a poverty-stricken outpost in the North Atlantic into one of the country’s wealthiest regions, progress has come at a high price — one that is only now becoming clear.

Last month, the Portuguese Finance Ministry ordered an investigation of Madeira’s accounts after unearthing what it called “a grave irregularity” — €1.1 billion, or $1.5 billion, of debt that had been accumulated since 2008 but not accounted for.

The discovery of yet more debt — equal to 0.3 percent of Portugal’s gross domestic product — complicates life for the national government as it struggles to meet financial targets agreed to last May with international creditors in return for a €78 billion bailout.